Lecture 1: Scalar and Vector Fields Different Types of Integrals: Volume, Line and Surface Lecture 2: Examples and Calculations of Surface Integrals Learning Goals: 1. describe what a vector field is and explain what information can be extracted from its magnitude and direction. Compare and contrast with a scalar field. 2. compute the surface integral of a scalar field or a vector field across a given surface by using an appropriate parametrization of the surface, and give a physical/geometrical interpretation of the result. For vector fields, relate the surface integral of the field to the concept of flux, and give a physical interpretation of flux. Lecture 3 Introduction to Divergence, Definition and Examples 3. Compute the flux of a vector field across a closed surface and give a physical interpretation of the result when the field represents, for example, the flow of a gas, or the electric field of a point charge. Lecture 4 Introduction to Gauss' Theorem Lecture 5 Continuation of Gauss' Theorem Lecture 6 Differential Operator Form of Divergence - Examples Lecture 7 Line Integrals - Relation to Potential Energy conservative fields?? Lecture 8 Explain what an oriented simple closed curve is, and give the mathematical definition of the circulation of a vector field around such a curve; explain how the circulation of the vector field relates to the tangential component of the field to the curve, and compute the circulation of a given vector field around a chosen curve. Lecture 9 Give a physical/geometrical interpretation and a mathematical definition of the curl of a vector field, and be able to compute the curl for a given field. Lecture 10 Apply Stokes' theorem to relate the surface integral of a vector field across a smooth surface to the line integral of the field along the boundary curve of the surface. Recognize when Stokes' theorem can be applied and use it to solve line integral problems. Divergence-free and curl-free fields Show that the curl of a scalar field with continuous second order partial derivatives is always zero. Explain why the line integral of a conservative field along a simple closed line and the curl of the field (show ???) are always zero; show that if a vector field has zero curl, the field is conservative. Use these facts to determine whether a given field is conservative. Given a conservative field, find an expression for its (scalar) potential function. Show that when the divergence of a vector field is zero, the vector field can be expressed as the curl of another field, which is called vector potential field.