This page includes six video lectures and links to associated lecture notes. http://ocw.mit.edu/resources/res-18-007-calculus-revisited-multivariable-calculus-fall-2011/part-iii 2013-01-22T06:47:37+05:00 MIT OpenCourseWare http://ocw.mit.edu en-US Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Video Description:</strong> Herb Gross describes n-dimensional vector spaces, relating definitions to the concept of a mathematical structure. Also covered: n-tuples in n-dimensional space; Structure of&nbsp;n-dimensional vector spaces; Definition of distance between two&nbsp;n-tuples; Limits of real-valued functions of several real variables.</p> <p><strong>Instructor/speaker:</strong> Prof. Herbert Gross</p>Keywords: n-tuples, Structure of n-Dimensional Vector Spaces<br><br>Thumbnail - <a href= http://img.youtube.com/vi/sSuZn6KHLnU/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.18-007/MITRES_18-007_Part3_lec1_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/part-iii-partial-derivatives/id491034051?i=108768199>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/sSuZn6KHLnU>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-18-007-calculus-revisited-multivariable-calculus-fall-2011/part-iii/lecture-1-n-dimensional-vector-spaces Gross, Herbert 2012-03-09T14:11:48+05:00 en-US n-tuples Structure of n-Dimensional Vector Spaces MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Video Description:</strong> Herb Gross introduces us to the traditional Calculus of Several Variables. He defines and explains the properties of partial derivatives and shows how to draw a graph of a function of several variables. He finds the normal vector (using the cross product) and the tangent plane at a point in terms of partial derivatives. Finally, Prof. Gross shows the change of a function w in the tangent plane as an approximation to the differential of w.</p> <p><strong>Instructor/speaker:</strong> Prof. Herbert Gross</p>Keywords: Partial Derivatives, Tangent Plane<br><br>Thumbnail - <a href= http://img.youtube.com/vi/UGKL1wHouho/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.18-007/MITRES_18-007_Part3_lec2_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/part-iii-partial-derivatives/id491034051?i=108768101>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/UGKL1wHouho>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-18-007-calculus-revisited-multivariable-calculus-fall-2011/part-iii/lecture-2-calculus-of-several-variables Gross, Herbert 2012-03-09T14:11:48+05:00 en-US Partial Derivatives Tangent Plane MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Video Description:</strong> Herb Gross defines the directional derivative and demonstrates how to calculate it, emphasizing the importance of this topic in the study of Calculus of Several Variables. He also covers the definition of a gradient vector.</p> <p><strong>Instructor/speaker:</strong> Prof. Herbert Gross</p>Keywords: Gradient Vector<br><br>Thumbnail - <a href= http://img.youtube.com/vi/SFB2Fxel6iM/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.18-007/MITRES_18-007_Part3_lec3_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/part-iii-partial-derivatives/id491034051?i=108768193>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/SFB2Fxel6iM>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-18-007-calculus-revisited-multivariable-calculus-fall-2011/part-iii/lecture-3-directional-derivatives Gross, Herbert 2012-03-09T14:11:48+05:00 en-US Gradient Vector MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Video Description:</strong> Herb Gross shows examples of the chain rule for several variables and develops a proof of the chain rule. He also explains how the chain rule works with higher order partial derivatives and mixed partial derivatives.</p> <p><strong>Instructor/speaker:</strong> Prof. Herbert Gross</p>Keywords: Higher Order Derivatives<br><br>Thumbnail - <a href= http://img.youtube.com/vi/JSs_dqq2uWo/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.18-007/MITRES_18-007_Part3_lec4_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/part-iii-partial-derivatives/id491034051?i=108768198>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/JSs_dqq2uWo>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-18-007-calculus-revisited-multivariable-calculus-fall-2011/part-iii/lecture-4-the-chain-rule Gross, Herbert 2012-03-09T14:11:48+05:00 en-US Higher Order Derivatives MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Video Description:</strong> Herb Gross show how the chain rule is involved in finding some integrals involving parameters. He computes the derivatives of integrals with constant limits, as well as derivatives of integrals with variable limits of integration (chain rule).</p> <p><strong>Instructor/speaker:</strong> Prof. Herbert Gross</p>Keywords: Variable Limits of Integration<br><br>Thumbnail - <a href= http://img.youtube.com/vi/__NxbJzEMCU/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.18-007/MITRES_18-007_Part3_lec5_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/part-iii-partial-derivatives/id491034051?i=108768203>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/__NxbJzEMCU>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-18-007-calculus-revisited-multivariable-calculus-fall-2011/part-iii/lecture-5-integrals-involving-parameters Gross, Herbert 2012-03-09T14:11:48+05:00 en-US Variable Limits of Integration MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm <p><strong>Video Description:</strong> Herb Gross explains the necessary and sufficient condition for an expression of the form Mdx + Ndy to be an exact differential.</p> <p><strong>Instructor/speaker:</strong> Prof. Herbert Gross</p>Keywords: Exact Differentials<br><br>Thumbnail - <a href= http://img.youtube.com/vi/Rvnv3bPDCs8/default.jpg>JPG (YouTube)</a><br>Video - download: <a href= http://www.archive.org/download/MITRES.18-007/MITRES_18-007_Part3_lec6_300k.mp4>Internet Archive (MP4)</a><br>Video - download: <a href= http://itunes.apple.com/us/itunes-u/part-iii-partial-derivatives/id491034051?i=108768109>iTunes U (MP4)</a><br>Video - stream: <a href= http://www.youtube.com/v/Rvnv3bPDCs8>YouTube </a><br><br><a href= 'http://ocw.mit.edu/terms/'>(CC BY-NC-SA)</a><br><br> http://ocw.mit.edu/resources/res-18-007-calculus-revisited-multivariable-calculus-fall-2011/part-iii/lecture-6-exact-differentials Gross, Herbert 2012-03-09T14:11:48+05:00 en-US Exact Differentials MIT OpenCourseWare http://ocw.mit.edu Content within individual OCW courses is (c) by the individual authors unless otherwise noted. MIT OpenCourseWare materials are licensed by the Massachusetts Institute of Technology under a Creative Commons License (Attribution-NonCommercial-ShareAlike). For further information see http://ocw.mit.edu/terms/index.htm