[{"run_id":"20260906T231458-5fdd2fff","semester":"1272","observed_at":"2026-09-06 23:14:58.172943+00:00","record_version_id":"6c6ee35a6f522e50e13176157a431c1adef588d7bc836dfaed1411f9efa2b3ef","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"20260907T155543-ce3781c4","semester":"1272","observed_at":"2026-09-07 15:55:43.033547+00:00","record_version_id":"6c6ee35a6f522e50e13176157a431c1adef588d7bc836dfaed1411f9efa2b3ef","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"20261014T203208-0851c9a1","semester":"1274","observed_at":"2026-10-14 17:50:09.728220+00:00","record_version_id":"e3749304d1339790e0c8b21adb106b854b1da9a9022ba8260dd6d0c9dbd5f860","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"45deb288dfed2b6b3932c2468efbd64c78525750f91c294c755d6a0e8311f799","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[{\"catalog_year\":\"2026-2027\",\"observed_at\":\"2026-10-14T17:51:41.022582+00:00\",\"source\":\"catalog\",\"source_url\":\"https://guide.wisc.edu/courses/math/\",\"term\":null,\"text\":\"Use the techniques of multilinear algebra and apply them to differential forms in Euclidean space and on abstract differentiable manifolds. This includes being able to compute with the exterior derivative.\"},{\"catalog_year\":\"2026-2027\",\"observed_at\":\"2026-10-14T17:51:41.022582+00:00\",\"source\":\"catalog\",\"source_url\":\"https://guide.wisc.edu/courses/math/\",\"term\":null,\"text\":\"Identify and/or construct differentiable manifolds. This includes being able to identify spaces as differentiable manifolds and to endow abstract topological spaces with differentiable manifold structures.\"},{\"catalog_year\":\"2026-2027\",\"observed_at\":\"2026-10-14T17:51:41.022582+00:00\",\"source\":\"catalog\",\"source_url\":\"https://guide.wisc.edu/courses/math/\",\"term\":null,\"text\":\"Extend calculus concepts from Euclidean space in order to apply them to abstract differentiable manifolds. This includes being able to identify and work with differentiable maps between differentiable manifolds in computations and proofs\"},{\"catalog_year\":\"2026-2027\",\"observed_at\":\"2026-10-14T17:51:41.022582+00:00\",\"source\":\"catalog\",\"source_url\":\"https://guide.wisc.edu/courses/math/\",\"term\":null,\"text\":\"Use the essential definitions and theorems from calculus on manifolds in the context of computations and proofs. This includes taking Lie derivatives of vector fields on differentiable manifolds, computing integrals of curves, integrating differential forms, and Stokes’ theorem.\"},{\"catalog_year\":\"2026-2027\",\"observed_at\":\"2026-10-14T17:51:41.022582+00:00\",\"source\":\"catalog\",\"source_url\":\"https://guide.wisc.edu/courses/math/\",\"term\":null,\"text\":\"Identify applications of course content in current areas of research.\"}]"},{"run_id":"legacy-03b0bbb9723907cf3d1b41cf0bb34837dc398e42","semester":"1262","observed_at":"2025-04-16 05:20:33.104844+00:00","record_version_id":"c2687ecf81e3960a23b26876057aedf1f87d6b5435e2f8421144b9569b091014","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-0837245627eb714ae6bb0375f7abe8f41042011f","semester":"1262","observed_at":"2025-04-16 05:20:33.104844+00:00","record_version_id":"c2687ecf81e3960a23b26876057aedf1f87d6b5435e2f8421144b9569b091014","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-08c92e359bbbc110b0778b7e0b15d6f40000e6b1","semester":"1262","observed_at":"2025-08-21 07:05:14.730696+00:00","record_version_id":"05b3c13699ea4565e0a716909a5a9044eebac34c6d75077edaef19caba339787","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-0944d76440dc778fbb89626058c10451009ce9c6","semester":"1262","observed_at":"2025-05-21 03:36:18.194118+00:00","record_version_id":"ed4f6d67467aafb10795d690f58514e6c0208485cceebc9d7a59278cee62fc1f","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-0a7f4bce29c816ef4d3d71b4eb8a2a3e15e33cc1","semester":"1262","observed_at":"2025-06-18 20:04:02.695444+00:00","record_version_id":"4c2dbd65a0d11ad4daa9ac9965d98e5f06fcf1c5e8b76648b55dfedf4c33e8d2","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-0ab917f827a7de22ebdc1087114d377a31c4687b","semester":"1262","observed_at":"2025-05-22 02:11:02.531191+00:00","record_version_id":"d6b3a06e8f1fa54a3bad779f05d8bae1f9f8efdc4059f85681ffe74632cad126","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-0c40ecab4b6db9333d1f18aef816e3ee659e300c","semester":"1262","observed_at":"2025-05-25 18:02:34.797617+00:00","record_version_id":"a3173c6ef8129e6ce78f6015b4153e9219d18405d73175fd7ea592e289e3ec30","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-0cb4d29a588ec7f11c7e00b7cf5dc3a0d23d0349","semester":"1266","observed_at":"2026-01-22 01:00:26.555427+00:00","record_version_id":"fc3f6295fe9b71a7a82aca358e5ec7d6a244fba23405607c8a3c89dc407fc4c1","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-11c30ec39e91498cbfcf3a923c53a4b03e387280","semester":"1262","observed_at":"2025-07-20 04:58:26.847117+00:00","record_version_id":"e9c1377cd31c51011583885eb8d8182a4df077fd2b036438d078189b689f6782","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-13cd500edf2587e9a3428f326253738fbea965ea","semester":"1264","observed_at":"2025-12-14 04:50:04.653658+00:00","record_version_id":"0824f0967ed1f422a43ad8deef12ddfca255112c9dc0de4ba34f7549e3ccf0ff","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-1da08b2de39f05e6bc877c189fc2fd6eeda563f5","semester":"1262","observed_at":"2025-07-01 00:03:56.707251+00:00","record_version_id":"3f2168989944fbd7a820eb452c8dca772daab125576c2dd968ee27cf67db2549","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-2031af47efe133b0e7877f433286ff07385afc41","semester":"1262","observed_at":"2025-07-02 20:16:45.749584+00:00","record_version_id":"883002fc95a93823779d08186b1bfaf2b9a0841cd13fc0d81e7b4352272b31e7","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-238eba6f32fbc30407f616dd0628ff02a2a824dd","semester":"1262","observed_at":"2025-08-17 05:15:07.788352+00:00","record_version_id":"6c2327696c30140fbd9734b618c00104b18c87077556aeba9cd5b576560a2b99","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-25f0ff56bc4d80f8a8afa1a0bcce8dae6fddb5e5","semester":"1262","observed_at":"2025-05-25 17:20:25.942090+00:00","record_version_id":"7c9cab84dc326cf106cb28b9b14625285bf57c761bae222d1ea1e756d3933d69","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-2923f91826ce253b7aeefa9f75b4a39fe4c80ff0","semester":"1262","observed_at":"2025-08-10 05:10:25.938416+00:00","record_version_id":"bb28aee540a9f5bcbb09ed236bed3db150bebb85103b13bcd42115d1657b6476","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-2a05214a4023e0157fa6f7d188c85308f714176c","semester":"1262","observed_at":"2025-09-07 04:56:38.255315+00:00","record_version_id":"320d64f20a6e4eb50b2f45f51c0459a04d1b52e15d6d2cf9f6bd742c359e10c3","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-2b1b8e39131de92b94620b25c5bbb25690efed4c","semester":"1262","observed_at":"2025-06-01 07:54:53.913454+00:00","record_version_id":"9ccfc28ca83e5861b8c4b701045dd20b026d0c8a470e8067bf149ac6f0b145ef","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-33a1225644e5a5855d60f66e65413946e72fedf4","semester":"1262","observed_at":"2025-06-22 01:52:05.152650+00:00","record_version_id":"810e3cce8489374f47ef7069195f5445d18f6b189a4176629ffebad531200ea3","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-3b89c83dbf5208e13756184de1f2b2d0c09af35f","semester":"1262","observed_at":"2025-06-30 06:40:10.235522+00:00","record_version_id":"3f3c2d2fceb89cad0b05287c2cea80b66a123a2c05e0759e3406e51aaaea33ff","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-3c3eebf7f35c39364b445a2fa5a112ce28dbb4a1","semester":"1262","observed_at":"2025-06-01 17:03:59.062985+00:00","record_version_id":"5a724ed8c62680466b847b260cdebc07e2f2386099f2954abad97e28773f6db7","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-3fa4dd71391f1c9cf4ab2216680bbc428689f92d","semester":"1262","observed_at":"2025-05-25 01:15:58.180530+00:00","record_version_id":"ce1e9b6f1a85f4819f6913722069fbfea31e0c761c10d87c51d6677c4daf051d","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-4040aa216da722a34483f68fc1e129ea571ae08d","semester":"1262","observed_at":"2025-06-01 07:27:39.695506+00:00","record_version_id":"263ac4efaf7ab3fd7e40d7fb3a10ad81eb92e2b429c0ef1cfca386f884e8149d","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-454d53cd690294459f8ad0793cc3ed562060e56c","semester":"1262","observed_at":"2025-05-25 01:34:00.536004+00:00","record_version_id":"5089d66127b4467221ed64c094e39547c09cd4ec3f26d86487d8eca43cb821ae","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-486f091e051fdb4ccdb80ee7b4fb0a5814510324","semester":"1264","observed_at":"2025-11-16 05:32:31.273726+00:00","record_version_id":"d7a80cf3ba13fcfbe132e5fbb9d5854336db5685cbbd53bc60bd9c8c5d940d26","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-50393cfc5fd5598d4f4731ef1919f2ff53af07d9","semester":"1262","observed_at":"2025-06-30 06:05:57.438846+00:00","record_version_id":"557a426d46a235a62da7de7c51542fb46db7028c2e591a8f4247a027be73c0ad","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-50f6947786b208c800db8bbd0cec84a8f0f187d2","semester":"1262","observed_at":"2025-06-02 03:49:42.915428+00:00","record_version_id":"f6b0261769e18c27d1ab0a3c806d9a22eb86179622f95afa1294df2dd9cb5ea2","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-51d86ed4082dced3803c295419b0ec8440942016","semester":"1264","observed_at":"2025-11-12 04:50:41.141307+00:00","record_version_id":"3d197cc9be4975931766bc319085557323b4ffc64b15db714db7f6993870da5c","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-53101bd1c10db0d00d094e4c9154d4c244ff23d9","semester":"1262","observed_at":"2025-10-05 04:58:05.366968+00:00","record_version_id":"3c01af8c422bea5265662ae56a2960d9b22c2139952c00bb587c5220140f26ec","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-58751151c564e11c5cb8aedd3a2d3527c22f73eb","semester":"1262","observed_at":"2025-06-01 06:59:01.536576+00:00","record_version_id":"341c05561078b537f0935aefc26e33526b88271162b4c09cf0290a88acd51129","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-5c4c8d3701a333ba4ddadef7d0b5dde8440fcdf0","semester":"1262","observed_at":"2025-06-01 01:57:51.256704+00:00","record_version_id":"f4c4bd1ae9e4986c529c1d65c06dea13f18db68a8b6bc8d0cf21c3c10f0b681c","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-5d20d9f81651d3c83fcaa2fd10870ce6e5d5a685","semester":"1262","observed_at":"2025-05-21 16:48:48.484741+00:00","record_version_id":"e406f6f95eaa872d1f2aaa3f1f36039105961cf7263d28da47b5c93d62773ec4","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-5feaebf2ab37051c86ad360cc0fa0d7e0e2ea295","semester":"1262","observed_at":"2025-05-24 15:58:23.378859+00:00","record_version_id":"9039648338fe9d5447c89d8a3c9bda2db0fb94f5fa56b4ccd912ee7b7491d853","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-60949965544fce098cc006354e201149005fb77b","semester":"1264","observed_at":"2025-10-26 05:07:14.077864+00:00","record_version_id":"778e32ea99f2351e0f59d9788d7a15cb190ba6fe6717dced89a7fdf8907342a0","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-639225ac3addf8522f1d30cb4fe26054ea7f8c49","semester":"1262","observed_at":"2025-06-01 08:14:42.497608+00:00","record_version_id":"abfa4ffd6a5db1ee388f8a1047f23b0715c5dc735d2939e1b0218663d254dd21","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-64093b17b3a9c67a41aa33eb37adb53ec38df714","semester":"1262","observed_at":"2025-05-22 02:11:02.531191+00:00","record_version_id":"72c6a11f9dee8648bf6b143bc41a17fa9037979649b85450aa26bebd91d6df73","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-6885b56099f55e08aa0205546369fb0381d50e13","semester":"1262","observed_at":"2025-07-13 04:59:22.367394+00:00","record_version_id":"cb63736099c162c4262e94d6631c0b339e9ee398ba450103f93828c2d4edbca9","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-69d4b2b16e8268f75b6390b3a0e2ce8bf73999da","semester":"1262","observed_at":"2025-06-30 03:21:29.925498+00:00","record_version_id":"74161984f7e7ee3da68a7eb7a86245a9ba530ea3552bd4525a09c9b43d98848e","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-6b29c3e0673b2f9a736d2363db5409e03d5b3ed4","semester":"1264","observed_at":"2025-12-03 12:33:41.418072+00:00","record_version_id":"4534608e1e1ec7be1c26fb9537721a9e0062da3844c8eda4adc0094f7cbb77d1","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-73259984748de484976ef2ad56ad7ab3d4cb4518","semester":"1262","observed_at":"2025-06-26 08:27:56.461216+00:00","record_version_id":"7962bf45da1546e933d6e19abc5e7b267a2947232f809e9cf09192acf32017fe","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-741602fbc6252c232a0b9e6bc1a9ae3101625f5a","semester":"1262","observed_at":"2025-06-30 22:49:32.522034+00:00","record_version_id":"a4083d00acfcf0200dfcfee171a3f13afc1227b55b98cf9d6988cd9d8e69111d","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-7774f45558276ec29bf626e58fffb4a27bdbb2a9","semester":"1264","observed_at":"2025-11-01 11:49:36.359151+00:00","record_version_id":"b4707852e4e07a87f5609637c25ecb6cb3c86514554b7e387b3638ee5cf9e7e2","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-7ab7c5e783fc8ebef8b8ee8cea019ee37a72ce74","semester":"1262","observed_at":"2025-09-28 04:58:07.340658+00:00","record_version_id":"764211d302505d82540c32132642e204c710c44553efcc3705edf722bea56763","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-7b2f22d83ea05055ab90dcb77dc802382cccc6f7","semester":"1262","observed_at":"2025-05-21 17:56:46.096741+00:00","record_version_id":"97fab2a4058ac9a0fc585ea7879971be375c6d1b1b6b9e1e7bcc81c14603c53c","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-7ca1175502be18eb51f7b016818a884a43fd4ebb","semester":"1262","observed_at":"2025-07-29 06:28:23.644795+00:00","record_version_id":"dcffbc6314ce05174ffeed769646861a5b22500cd95f172d94c62f53d6c38c00","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-7d8cee5e42602ad9ba5dc0322085af2fa0dffaac","semester":"1262","observed_at":"2025-05-22 01:15:19.291664+00:00","record_version_id":"d32a05296da033aa2338b12de9b602216cb5f114ae24fc3a96080b4e5105e7a8","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-81a0d6a7f668c98c59bd05fa988aa079c5ad1f68","semester":"1262","observed_at":"2025-05-31 07:49:50.419452+00:00","record_version_id":"1044373740c2f5b05757047c20943d09514c3eb6bbaf7ddbf5d9d6070142824b","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-82f31c08c746a214872494cde6ecaeb9330441b4","semester":"1262","observed_at":"2025-06-17 21:26:08.547156+00:00","record_version_id":"58d7703c49875576b62d48404352875917b225c39e3cc3690158be3888ebe626","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-84c248343c5a99ab42c5616c594812a5e3e31cbb","semester":"1262","observed_at":"2025-07-04 06:00:07.487827+00:00","record_version_id":"b44ba4a1b3e8b45745f855f00b8a7e568786ac5861480e66764c9a69c57eaa97","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-86334e7275e315853287e8580c35456c68df800b","semester":"1262","observed_at":"2025-05-22 06:53:03.336354+00:00","record_version_id":"0eb40f1430f36ca800119ee27ee58eb5655b475e2423e1170b4e0f9f81af13c0","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-8686dc1664f93a6c05f50321410961b39b12f91b","semester":"1262","observed_at":"2025-05-28 10:17:41.595894+00:00","record_version_id":"43b7a79b34020d4c2b414fef259fbd705b2a321d855b0b532d97a3e3ff805080","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-86d941366886c9a14a2581c04e20d0bdd25e1b02","semester":"1262","observed_at":"2025-08-31 04:54:14.656039+00:00","record_version_id":"a03bca68eff95f543f5ef25144c09dd4d70cdb2722d1041494a6ac9f2083f70d","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-8ba1d2e6cfbb1525b4f9f769ec164fb6ad0f237a","semester":"1262","observed_at":"2025-06-02 05:22:45.010398+00:00","record_version_id":"b9e498f4dc50c6b66aff2182ede6c9d0930fb0b0abdb110e55137a3dab5be613","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-90e5aa456689a3b222016c05c319bbcd0bddd453","semester":"1262","observed_at":"2025-05-25 02:00:10.037905+00:00","record_version_id":"e432b448861b478735be146fbea30cc9f9692a4115379d9e196fad57eaef6c09","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-92d960aa7e4cd6e4c9ac8b2e232092904038e9e4","semester":"1262","observed_at":"2025-06-08 01:50:46.451351+00:00","record_version_id":"b78629f60b63f8632ce579756748d1f3318b3962db707bba723bf4021032d70f","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-94002ae588a8f71eca6d64105f2e603e8587293b","semester":"1262","observed_at":"2025-05-21 09:01:16.021508+00:00","record_version_id":"69c0562ce5a1fa6219e5d515c0af1c419152cfd0bb85015d30932889dde5bbb3","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-9dd575e013c5c6e000ab3ad2d1288cc1dd9c96dc","semester":"1264","observed_at":"2025-12-21 04:47:46.589081+00:00","record_version_id":"48dc66f7c76bf02341841871e912bfde79367329b88e86bdcee73a27d9a85b79","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-a1ea26ed78cbd6e94bc3221e50a6ed9b6bf43871","semester":"1262","observed_at":"2025-05-25 09:16:55.180581+00:00","record_version_id":"972f347c0650075dfdb06ea95f1b5bf1a2b4ea81a931c39190ceb75eed4b1dc7","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-a76a3505853d5ea9c13bb98472b87361d469e2ee","semester":"1262","observed_at":"2025-05-28 18:03:41.793846+00:00","record_version_id":"3d5981737bf730dd226e3fe1fd9bccc4ca24c81cee9f5bc2e43ce0cf8d4befb9","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-b285d591f6acf3aa1910e02edc297664db86eb8c","semester":"1262","observed_at":"2025-06-10 22:48:22.460295+00:00","record_version_id":"d2658cad613914fe3ad45e3070a9f50ed67cce8332d9d8973a5c9ce2f17de754","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-b79af756e4f00ba98231feb859ed8e98239eb811","semester":"1262","observed_at":"2025-05-22 02:47:56.931581+00:00","record_version_id":"40409914073e3f10997a3d8c3c25c3e83c7e09ab7e34e32de67e035aa4c83f3b","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-b89e05b94730c6471439082d9279c0a834a4aeec","semester":"1262","observed_at":"2025-08-26 20:25:42.325239+00:00","record_version_id":"ac95e97be45015b8f5db37d8f724a0e842b3a44bf176e60f0f2ac8117b0fb957","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-b9279a12d1f2688df9e556183c94d41a89e431dc","semester":"1262","observed_at":"2025-07-06 01:49:47.203835+00:00","record_version_id":"e3c9b490baaa11569b65ff65743ef2994866e6bee06029ab7151f4bfc44733de","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-bbdc905e96cc681a5e9a8b02c795b682206e36d1","semester":"1262","observed_at":"2025-06-25 08:02:45.445730+00:00","record_version_id":"a58009bb62c352204e837ae59504fef599d670f282a739bb2c603ae79ebbb2a6","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-bde225b5243a13d81827aa173e39c78a652b54a0","semester":"1262","observed_at":"2025-05-23 07:22:33.219345+00:00","record_version_id":"bb142e3270bd555c45c7ab59867293b474291d7a9bd84fa50bfd8afd269f7b18","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-c1fac4cbfde737f4f4ac02cdf15e6f5a9b5b676d","semester":"1262","observed_at":"2025-05-22 01:15:19.291664+00:00","record_version_id":"d32a05296da033aa2338b12de9b602216cb5f114ae24fc3a96080b4e5105e7a8","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-c6b62be45f8a13cba109ef2a7741b9cd9a6b8272","semester":"1262","observed_at":"2025-08-24 08:48:09.035112+00:00","record_version_id":"dd95d195f296c20146b4007ac9c583e8ed28e662fcf2a995bc459f2d544a4321","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-cc83f62cb6db61f39320916d819ad48ce55cd96d","semester":"1262","observed_at":"2025-05-22 00:45:35.735230+00:00","record_version_id":"78f6cb6f45dbfe762240296a5f499b3439381869657b1a9a131ed66333b0628f","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-d33818a01b6b4c5e2902e704da6e5b713742d4dc","semester":"1262","observed_at":"2025-10-13 10:46:24.251475+00:00","record_version_id":"f4d1fb5d5d8a42cb81f5d1992a79effb257e5937925cc616c2a4c68b86318e52","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-d3c17db9832d3064916046caf891dd23f10b1bb9","semester":"1262","observed_at":"2025-08-26 11:30:42.288639+00:00","record_version_id":"d27279e08423fb21c091bebc86854eab241e71513be55ee9f752a0e90e83f92c","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-d4ee226dc44268e9b6767fb9f5f20343322f494b","semester":"1262","observed_at":"2025-06-15 01:52:23.911697+00:00","record_version_id":"dba3300da4c5a17ec77d74f34d146f1f0d2557c1e34751ea152361ee5f816c9e","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-d8f9fc7f136e2033556d9d65bbdbdf33425dbdc9","semester":"1262","observed_at":"2025-05-28 16:48:12.400438+00:00","record_version_id":"71cf0358f320daae2ce285e31421081398e35928c6425038e8943e505fc5c829","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-dbbc9739c616c3846b2e796e00a36fa41225f7d8","semester":"1262","observed_at":"2025-06-01 17:47:19.325144+00:00","record_version_id":"c374982ecf66557d6e318b1f9ee84693bd61252a0c8f8aabd1f35bbb256195cd","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-dbe5f2d24b48c3177297a3d69c553cc9d1194bac","semester":"1262","observed_at":"2025-09-14 05:00:27.366070+00:00","record_version_id":"c74de2eb2e5a8d117f08c57f66c4799d017cb8d8d4449008d00a7355132fe600","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-dd357dc2758e9e282c30988a527ce37de92593b9","semester":"1264","observed_at":"2025-10-19 01:39:41.951404+00:00","record_version_id":"3e9d44d050b59bfd5f43189629a2a53cd51b8acca5c19f4238e0fe242117baea","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-de74de559e409a3fe668685e28043b888d2e6841","semester":"1262","observed_at":"2025-06-27 07:38:50.882897+00:00","record_version_id":"65118acb66ef3671a54b854d80d3d8a54726d36fadcdf5a206e4ab927803ac9f","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-deca7188cf536ade3203cd88b52a45e92a059617","semester":"1262","observed_at":"2025-06-27 00:37:46.087071+00:00","record_version_id":"74c6e045dc12486b17865159fee594fc03267390d7889a77edafe90e19fc623b","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-df60fa9ac3931c5612b0fa641b7a6f5fbec86fab","semester":"1262","observed_at":"2025-05-24 06:36:37.554678+00:00","record_version_id":"f18c9894762a622a0787d2af5792b38ccc25e4c66c4d52419de299cb3e191c53","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-e339da2d849aaf8bb4dceca07a84728af23cf755","semester":"1262","observed_at":"2025-07-23 08:49:22.748925+00:00","record_version_id":"8dfecaf8e1ffec46c1a7ba8425a44069d42dd240987bdbaf00a9dfbf948a6def","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-e934b1784835f5d36ae9b7ef0992b11e72c39142","semester":"1262","observed_at":"2025-06-01 06:21:25.219533+00:00","record_version_id":"0a40b05115e65a4bf5789a17c97c3302a4abf65c66f3c585f3e84dac8e58233e","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-ef471690c082448deebd2687eab84b0b78813c36","semester":"1262","observed_at":"2025-06-29 01:47:56.317645+00:00","record_version_id":"a3cc49909b49eeb889902bed15d659dab2b9467621fbc04a1402c498ef6a87a9","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-efc4e8e2cd7c3f68e059a308990c00c77c7611eb","semester":"1264","observed_at":"2025-10-16 05:48:13.935141+00:00","record_version_id":"4a343928f2f785ae9b8b5d17ffe240ad0f7d1bdfe8a2b8310757519225a2bb34","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"},{"run_id":"legacy-f18b7749725097d41f69c60779ff38cf073b93b3","semester":"1262","observed_at":"2025-05-21 19:52:03.259353+00:00","record_version_id":"b83927629246c6c257759df3629aaa8e3e7942a3bd13af5a718b91d28df08488","course_id":"MATH 621","course_uid":"course_abe8dad788555fbfa877b029","catalog_version_id":"e6ba7cf69214fc5558d507dd8caf7f989a8c778fc283c3954893015357e583dd","course_number":621,"subjects":["MATH"],"title":"INTRODUCTION TO MANIFOLDS","description":"Integration in Euclidean spaces, change of variables, multilinear algebra, vector fields and differential forms, manifolds and tangent spaces, integration on manifolds, Stokes theorem. Classical vector analysis. Cauchy integral theorem.","requirements_text":"MATH 522, (MATH 521and561), graduate/professional standing, or declared in Mathematics VISP","official_learning_outcomes_json":"[]"}]