Learning expectations for the final
- Be able to find parameterizations of basic curves such as line segments, and circles.
- Be able to find a parameterization of a curve given as the intersection of two surfaces.
- Be able to find the intersection (if it exists) of two parameterized curves.
- Be able to take the derivative of a vector valued function and have facility with the product rules and the chain rule. Use properties of cross and dot product to simplify expressions involving derivatives
- Be able to find the tangent lines to a curve at a point. Be able to solve more involved problems involving tangent lines (e.g. "at what point on the curve is the tangent line normal to the plane...").
- Be able to compute the arc length of a curve.
- Be able to reparemeterize a curve by arclength.
- Know the conceptual definition of curvature and be able to compute curvature from a parameterization.
- Know the conceptual definition of curvature as well as both computationally useful formulas.
- Know the definition of torsion and its meaning.
- Be able to compute the osculating plane and osculating circle to a curve.
- Know Newton's law of motion and be able to determine the motion of a particle under the influence of a force in simple cases. In particular, you should know the cases we did in class and in the book (thrown baseball, skier on a hill).
- Know the tangential and normal components of acceration.
- Be able to state Kepler's three laws.
- Know the proof of the conservation of angular momentum (done in class for planetary motion), well enough to derive conservation laws in similar circumstances.
- Be able to draw a plot of a simple vector field.
- Know the definition of conservative vector field well enough to work with it in simple cases.
- Be able to determine if a vector field F(x,y) is conservative, and if so be able to find a potential function f(x,y).
- Know well the logical relationships between the properties
- F(x,y) = P(x,y)i +Q(x,y)j is conservative
- F(x,y) = P(x,y)i +Q(x,y)j satisfies the path independence property
- F(x,y) = P(x,y)i +Q(x,y)j integrates to zero over all loops
- P_{y}(x,y)=Q_{x}(x,y)
- P_{y}(x,y)=Q_{x}(x,y) and the domain is simply connected
- Know Green's theorem.
- Be able to use Green's theorem to compute area in terms of a line integral.
- Be able to use Green's theorem to compute line integrals: primarily around closed loops, but also possibly for non-closed curves (where you must add a curve or curves to make a loop).
- Apply Green's theorem to domains with both interior and exterior boundaries.
- Know the definitions of divergence and curl and their basic properties.
- Be able to derive various product formulas for divergence, gradient, and curl.
- Be able to find parameterizations of surfaces, including surfaces contained in a plane and parts of spheres.
- Know and understand the standard parameterization of the sphere
- Use a surface integral to compute surface area of a parameterized surface.
- Be able to compute the integral of a function over a surfaces.
- Be able to compute the integral of a function over a surface using a parameterization of a surface.
- Be able to check whether or not a parameterization of a surface is compatible with a given orientation of the surface.
- Be able to find the boundary of a surface, and figure out its induced orientation.
- Be able to compute the flux of a vector field through an oriented surface using a parameterization of the surface (including getting the orientation right!).
- Be able to compute a flux integral without using a parameterization in the special situations where the relationship between the vector field and the normal vector can be determined without a parameterization.
- Know Stoke's theorem and use it to compute work integrals around loops.
- Be able to use the divergence theorem to compute flux through a closed surface.
- Be able to apply the divergence theorem to solid regions with cavities
- Be able to complete a non-closed surface into a closed surface by adding a surface. Do this to evauate a difficult flux integral involving a complicated vector field whose divergence is simple.
- Be familiar with the vector field F(x,y,z) =r^{ -3} r where r = xi+yj+zk and r=|r|. This field has divergence 0, but is not defined at the origin. Moreover, the flux of this vector field through a sphere centered at the origin (and any radius), oriented outward, is 4 π . We used this and the divergence theorem to study the flux of F through any closed surface.