## Learning expectations for the Final Exam

The exam will cover material from the first and second midterms. Please note that the new material will be well represented since you have not been tested on it yet.

You should be able to:

• Match plots of curves with vector valued functions.

• Find parameterizations of curves which have been given geometrically (e.g. lines, circles, intersections of two surfaces).

• Find derivatives of expressions involving vector valued functions, dot products, and cross products. Simplify such expressions using geometric properties of cross product and dot product.

• Find the arclength of a curve.

• Reparameterize a curve by arclength.

• Know three formulas for the curvature of a curve and be able to apply them to compute curvature and/or answer conceptual questions.

• Compute the unit tangent vector, normal vector, and binormal vector to a curve at a point.

• Find the osculating plane and circle to a curve at a point.

• Find position the position vector of a curve when given the acceleration vector; apply Newton's law of motion.

• Find the tangential and normal components of acceleration. Understand the intuitive meaning of each.

• Know the definition of a conservative vector field and a potential function.

• Compute integrals of vector fields over curves (work integrals) in both vector and differential form notation

• Determine if a vector field is conservative.

• Find the potential function of a conservative vector field.

• Use the fundamental theorem of line integrals to compute work integrals.

• Carefully discuss the relationships between various properties of vector fields: being conservative, satisfying the path independence property, integrating to zero around closed loops, having their curve be zero, having a non-simply-connected domain.

• Use green's theorem to compute work integrals around closed loops. Apply Green's theorem to domains with holes. Use it to compute the area.

• Compute the divergence and curl of vector fields and the gradient of functions.

• Derive simple identities involving div, grad, and curl, for example, product formulas.

## Material since the last midterm:

• Parameterize a variety of surfaces, in particular pieces of spheres, cylinders, planes, surfaces of revolution, and graphs of functions.

• Match parameterizations with surfaces.

• Compute the surface area of a parameterized surface.

• Compute the integral of a function over a parameterized surface

• Compute the integral of a vector field over a parameterized surface (a flux integral) .

• Find the normal vector field of a surface corresponding to a given orientation.

• Find the induced orientation on a curve which is the boundary of an oriented surface. Conversely, find the orientation on a surface which induces a given orientation on its boundary.

• Apply Stoke's theorem to compute a work integral over closed curve.

• Apply the divergence theorem to compute the flux through a closed surface.

• When appropriate, be able to use some of the "advanced tricks" with Stoke's theorem, Green's theorem, and the divergence theorem. For example: compute a line integral by closing off a curve into a loop and applying Stoke's or Green's theorem, or compute a flux integral by closing off the surface and applying the divergence theorem.