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This is an advanced course in multivariate calculus. This course covers the same subjects (roughly) as MA 18 and MA 20, but is on a higher level, and is more difficult.

Topics will include:

  1. Differentiation

    • Brief survey of Euclidean geometry, scalar and vector products.
    • Multivariate functions: graphical representation (surfaces), continuity.
    • Differentiation in two and three dimensions: partial derivatives, directional derivatives.
    • Gradients, tangent lines and planes.
    • Extremal problems.
    • Lagrange multipliers and constraints.
    • Higher order derivatives and Taylor's theorem.
    • The implicit function theorem, the inverse function theorem.

  2. Integration

    • Brief survey of one dimensional integration.
    • Integration in two dimensions: Cartesian, polar.
    • Fubini's theorem.
    • Integration in three dimensions: Cartesian, cylindrical, spherical.
    • Change of variables: the Jacobian.
    • Geometrical applications: solid volumes, surface area, center of mass.

  3. Vector analysis

    • Vector valued functions.
    • The divergence and the curl of a vector field.
    • Line integrals in two and three dimensions.
    • Green's theorem (in two dimensions).
    • Surface integrals.
    • Divergence theorem (Gauss' theorem).
    • Stokes' theorem.

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