Multivariable Calculus

MIT , Prof.Denis Auroux

Double integrals

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Course Description

Dot product – Determinants – cross product – Matrices – inverse matrices – Square systems – equations of planes – Parametric equations for lines and curves – Velocity, acceleration – Keplers second law – Review – Level curves – partial derivatives – tangent plane approximation – Max-min problems – least squares – Second derivative test; boundaries and infinity – Differentials; chain rule – Gradient; directional derivative; tangent plane – Lagrange multipliers – Non-independent variables – partial differential equations – Double integrals – Double integrals in polar coordinates – applications

Change of variables – Vector fields and line integrals in the plane – Path independence and conservative fields – Gradient fields and potential functions – Greens theorem – Flux; normal form of Greens theorem – Simply connected regions -Triple integrals in rectangular and cylindrical coordinates – Spherical coordinates; surface area – Vector fields in 3D – surface integrals and flux – Divergence theorem – Line integrals in space, curl, exactness and potentials – Stokes theorem -Topological considerations – Maxwells equations – Final review

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