## Syllabus

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.