Test4reivew - calculus 3 test 4 review problems PDF

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calculus 3 test 4 review problems...

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Name/Row/Seat Number: R. Vogt 1. (20 points) Suppose f (x, y, z) = xyz R (a) (10 points) What is C f (x, y, z)dS, where C is the line segment that starts at (0, 0, 0) and goes to (1, 1, 1) R (b) (10 points) What is C f (x, y, z)dS, where C is the line segment that starts at (1, 1, 1) and goes to (0, 0, 0)  2 x 2. (20 points) Suppose f (x, y) = 2 (Just another way to write a vector field) y R (a) (20 points) Calculate C f (x, y)dr, where C is the unit circle. 3. (20 points) Suppose we have a vector function F = (x2 , y2 )

(a) (10 points) Show F is a conservative vector field. R (b) (10 points) What is C F dr where C is the unit circle. RR 4. (20 points) (a) (20 points) Find A(S) = 1dS, where the surface of interest is a unit sphere (hint S the parametric surface is r(u, v) = (sin(u)cos(v), sin(u)sin(v), cos(u)), 0 ≤ u ≤ π, 0 ≤ v ≤ 2π (How would this change if instead of a sphere with radius 1, it was a general radius ρ? RR 5. (20 points) (a) (20 points) Suppose we have the vector field F = (x, y, z). Calculate F · dS where S S is again the sphere with radius 1. (See previous question for its parametric representation). 6. (40 points) Recall Optimization problems (Lagrange Multipliers). When we found a min or max of the function it was a point in a vector space (we were optimizing over vector space). Lets introduce a new kind of optimization problem, lets optimize over function space (we call these problems Calculus Of Variations). See the follow example.

miny

Z

1 0

q

 2 1 + y ′ dx s.t y(0) = 0 y(1) = 1

English: We wish to find a single variable function function y which minimizes the objective function (the functions arc length from 0 to 1). Furthermore we require that this function also satisfies y(0)=0 and y(1)=1. We can do some math - and write a differential equation whose solution will lead us to the optimal y. We call this differential equation the Euler Lagrange Equation associated with this single variable calculus of variation problem. We write the differential equation in terms of the integrand. The general Euler Lagrange Equation for a general integrand F is   ∂F d ∂F − =0 ∂y dx ∂y ′ (a) (20 points) Write the Euler Lagrange equation for our specific case of F. Notice for us q down  2 ′ ′ F (y, y , x) = 1 + y .

(b) (20 points) Show that if y=x, then this causes the Lagrange Euler Equation to be satisfied.

English. The shortest curve you can draw between two points is a line. Interesting how optimizing over functions just comes down to solving differential equations? Neat. These also exist for partial differential equations too. They are fun! :)...