Final Exam - MATH 317 (2015T2)

Problem 1

(a). Compute and Simplify for and . Express Your Answer in terms of

(b). Compute

(c). Find the Length of the Curve for

(d). Find the Principal Unit Normal Vector to at

(e). Find the Curvature of at

Problem 2

Let be the surface obtained by revolving the curve , , around the -axis, where the orientation of has pointing toward the -axis.

(a). Draw a Picture of and Find a Parametrization of

(b). Compute the Integral

(c). Compute the Flux Integral where

Problem 3

Let be the parametrized curve given by , , and let .

(a). Compute and Simplify

(b). Compute the Work Integral

Problem 4

(a). Use Green's Theorem to Evaluate the line Integral $$\int_C \frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy$$

where is the arc of the parabola from to . (Hint: Green's theorem must be applied to a closed curve; note that the curve is not closed. You may use the fact that .)

(b). Use Green's Theorem to Evaluate the line Integral $$\int_C \frac{-y}{x^2+y^2}dx + \frac{x}{x^2+y^2}dy$$

where is the arc of the parabola from to . (Hint: Consider carefully the point in your analysis of the situation.)

(c). Is the Vector Field Conservative? provide a Reason for Your Answer Based on Your Answers to the Previous Parts of This Question

Problem 5

Consider the curve that is the intersection of the plane and the cylinder , and suppose is oriented so that it is traversed clockwise as seen from above. Let . Use Stokes' theorem to evaluate the line integral .

Problem 6

Let be the solid region between the plane and the paraboloid . Let

(a). Compute the Flux of outward through the Boundary of

(b). Let Be the part of the Paraboloid Lying below the Plane, Oriented so that Has a Positive Component. Compute the Flux of through

Problem 7

A particle moves along a curve with position vector given by for .

(a). Find the Velocity as a Function of

(b). Find the Speed as a Function of

(c). Find the Acceleration as a Function of

(d). Find the Curvature as a Function of

(e). Recall that the Decomposition of the Acceleration into Tangential and Normal Components is given by the Formula $$\vec{r}''(t) = \frac{d^2s}{dt^2}\hat{T}(t) + \kappa(t)\left(\frac{ds}{dt}\right)^2\hat{N}(t).$$

Use this formula and your answers to the previous parts of this question to find , the principal unit normal vector as a function of

(f). Find an Equation for the Osculating Plane at the point Corresponding to

(g). Find the Centre of the Osculating Circle at the point Corresponding to

Problem 8

Consider the following surfaces:

is the hemisphere given by with .
is the cylinder given by .
is the cone given by .

Consider the following parametrisations:

A. , ,

B. , ,

C. , ,

D. , ,

E. , ,

F. , ,

G. , ,

H. , ,

I. , ,

J. ,

K. ,

L. ,

For each of the following, choose all valid parametrisations from the list above.