Math 513 - Review Guide

*Everything* we covered in class is required for the final exam. However, here is *a list of basic topics, not to be missed*:

Ch.1

Operations with vectors (geometrically and using components).

Equations of a line (parametric and non-parametric).

Scalar product: using components, or using the cos.

Decomposing a vector into a component parallel to a vector plus a normal component.

Equations of a plane. Be able to write equations for a plane determined 1)by 1 point in the plane and the normal direction, 2) by 1 point and a line contained in the plane, 3) by two lines contained in the plane, 4) by 3 points contained in the plane.Be able to find the angle a) between a line and a plane, b) between two planes, c) between two lines.

Vector products: using components, the sine of the angle between the two vectors, the area of the parallelogram determined by two vectors, the area of the triangle determined by three points.

Ch.2

Curves given parametrically.

Tangent vectors. Unit tangent vector. Calculation of arc length. If a curve is parametrized by its arc length R(s) then the tangent vectors dR/ds are unit vectors.

Ch.3

Scalar fields: level lines and level surfaces (isotimic surfaces).

grad f (calculation, points in the direction of the fastest increase, it is orthogonal to the level surfaces of f, del notation, directional derivatives).

Flow lines: sketch, calculation.

Div: calculation, interpretation as source/sink, write using del notation

Curl: calculation, interpretation as the tendency to rotate a paddle wheel probe, del notation

The Laplacian: calculation, del notation

Linear approximations.

Be able to prove the vector identities (3.27)-(3.30) at page 147.

Ch.4

Line integrals: calculation.

Conservative fields, potential: definition, be able to calculate and use the potential, be able to state and use Theorems 4.1 (p.197), 4.2 (p. 201), 4.3 (p.405)

Surface integrals: calculation (using **R**xd**S**, or **R**x**n** dS, or dx dy/cos(gamma))

Volume integrals (calculation)

The Divergence Theorem.

Stokes' Theorem.