M5: Multivariable Calculus (2021-22)
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- Lecturer: Profile: Richard Earl
Course information
Course Term: Hilary
Course Lecture Information: 16 lectures
Course Overview:
In these lectures, students will be introduced to multi-dimensional vector calculus. They will be shown how to evaluate volume, surface and line integrals in three dimensions and how they are related via the Divergence Theorem and Stokes' Theorem - these are in essence higher dimensional versions of the Fundamental Theorem of Calculus.
Learning Outcomes:
Students will be able to perform calculations involving div, grad and curl, including appreciating their meanings physically and proving important identities. They will further have a geometric appreciation of three-dimensional space sufficient
to calculate standard and non-standard line, surface and volume integrals. In later integral theorems they will see deep relationships involving the differential operators.
to calculate standard and non-standard line, surface and volume integrals. In later integral theorems they will see deep relationships involving the differential operators.
Course Synopsis:
\(\bullet\) Multiple integrals: Two dimensions. Informal definition and evaluation by repeated integration; example over a rectangle; properties. General domains. Change of variables. Examples. [2]
\(\bullet\) Volume integrals: Jacobians for cylindrical and spherical polars, examples. [1.5]
\(\bullet\) Recap on surface and line integrals. Flux integrals including solid angle. Work integrals and conservative fields. [2]
\(\bullet\) Scalar and vector fields. Vector differential operators: divergence and curl; physical interpretation. Calculation. Identities. [2.5]
\(\bullet\) Divergence theorem. Example. Consequences: Green's 1st and second theorems. \(\int_V \nabla \phi dV = \int_{\delta V} \phi dS\).
\(\bullet\) Uniqueness of solutions of Poisson's equation. Derivation of heat equation. Divergence theorem in plane. [4]
\(\bullet\) Stokes's theorem. Examples. Consequences. The existence of potential for a conservative force. [2]
\(\bullet\) Gauss' Flux Theorem. Examples. Equivalence with Poisson's equation. [2]
\(\bullet\) Volume integrals: Jacobians for cylindrical and spherical polars, examples. [1.5]
\(\bullet\) Recap on surface and line integrals. Flux integrals including solid angle. Work integrals and conservative fields. [2]
\(\bullet\) Scalar and vector fields. Vector differential operators: divergence and curl; physical interpretation. Calculation. Identities. [2.5]
\(\bullet\) Divergence theorem. Example. Consequences: Green's 1st and second theorems. \(\int_V \nabla \phi dV = \int_{\delta V} \phi dS\).
\(\bullet\) Uniqueness of solutions of Poisson's equation. Derivation of heat equation. Divergence theorem in plane. [4]
\(\bullet\) Stokes's theorem. Examples. Consequences. The existence of potential for a conservative force. [2]
\(\bullet\) Gauss' Flux Theorem. Examples. Equivalence with Poisson's equation. [2]
Section outline
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This sheet covers the material from Lectures 1-2.
Multiple Planar Integrals.
Change of Variable
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This sheet covers the material from Lectures 3-4.
Multiple Integrals.
Change of Variable
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This sheet covers the material from Lectures 5-6.
Surface Integrals.
Flux Integrals.
Solid Angle.
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This sheet covers the material from Lectures 7-8.
Div, Grad and Curl.
Physical interpretation.
Identities.
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This sheet covers the material from Lectures 9-10.
Green's Theorem.
Divergence Theorem.
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This sheet covers the material from Lectures 11-12.
Divergence Theorem.
Examples.
Consequences.
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This sheet covers the material from Lectures 13-14.
Stokes' Theorem.
Examples.
Consequences.
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This sheet covers the material from Lectures 15-16.
Gravity.
Gauss' Flux Theorem.
Poisson's Equation.
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