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.