2. Below are four algebraic vector fields and four sketches of vector fields. Match them.

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1 Math 511: alc III - Practice Eam 3 1. State the meaning or definitions of the following terms: a) vector field, conservative vector field, potential function of a vector field, volume, length of a curve, work, surface area, flu integral b) curl and divergence of a vector field F, gradient of a function c) d) da or ds or e) dr da or,, ds or F or F S ds or d or dy or,, nds f) M ( d N( dy P( dz g) What does it mean when a line integral is independent of the path? h) State the Fundamental Theorem of Line Integrals. Make sure to know when it applies, and when it helps. i) State Green s Theorem. Make sure to know when it applies, and in what situation it helps. j) State Gauss Theorem. Make sure to know when it applies, and in what situation it helps.. Below are four algebraic vector fields and four sketches of vector fields. Match them. [A] [B] [] [D] (1) F ( y, () F (, (3) F ( 1, (4) F ( 1, y b) Below are two vector fields. Which one is clearly not conservative, and why? c) Say in the vector field [] above you integrate over a straight line from (0,-1) to (-1,0). Is the integral positive, negative, or zero? 3. Are the following statements true or false: a) If the divergence of a vector is zero, the vector field is conservative. b) If F ( is a conservative vector field then curl ( F) 0 c) If a line integral is independent of the path, then F dr 0 for every path d) If a vector field is conservative then F dr 0 for every closed path

2 e) da denotes the surface area of the region f) ds denotes the volume of the region g) an you apply the Fundamental Theorem of line integrals for the function y sin( cos( y )? h) an you apply the Fundamental Theorem of line integrals for the vector field F ( 6y 3,6 y 3y 7? i) an you apply Green s theorem for a curve, which is a straight line from (0,0,0) to (1,,3)? j) an you apply the Divergence theorem to the plane +y+z=1 over [-1, 1] [-1, 1]? 3 4. Suppose that F( y z, z, y is some vector field. a) Find div(f) b) Find curl(f) c) Find curl(curl(f)) d) Find div(curl(f)) 0 e) grad., div., and curl of the vector field if appropriate for, y, z grad =n/a, div = +y+z, curl = 0 f) grad., div., and curl of the vector field if appropriate for cos( y ) ycos( ),sin( ) sin(, yz grad = n/a, div =, curl = g) grad., div., and curl of the vector field if appropriate for z ln( y ) grad = 5. Decide which of the following vector fields are conservative. If a vector is conservative, find its potential function a) F ( b) F( e cos(, e sin( c) F ( sin(, cos 1 d) F( z, zy e) F ( 6y 3,6 y 3y 7

3 f = 3^ y^ ^3 + y^ f) F ( y sin(),3y (1 cos() g) F( 4y z, 6 z h) F( 4y z, 6yz, z 6. Evaluate the following integrals: a) cos( )da where is the triangular region bounded by y = 0, y = and = 1 b) ds, where S is the portion of the hemisphere y 9 y 5 that lies above the circle c) y 3zds where is a line segment given by r( t) t,t, 3t, 0 t 1 d) F dr where F ( and is the curve given by r ( t) 4 t,4t t, 0 t 3 e) yd dy where is a parabolic arc given by r ( t) t,1 t, 1 t 1 f) ( ds where S is the first-octant portion of the cylinder y z 9 between = 0 and = 4 S g) Find the flu of the vector field F ( z through the surface given by potion of the paraboloid z 4 y that lies above the y-plane. Note that this surface is not closed.

4 7. For the following line integrals there is a short-cut you can use to simplify your computations (but justify your shortcut by quoting the appropriate theorem) a) F dr where F( e cos(, e sin( and is the curve r ( t) cos( t),sin( t), 0 t b) yzd zdy ydz where is some smooth curve from (0,0,0) to (1,4,3) 3 c) F dr where F ( y 1,3 y 1 and is the upper half of the unit circle, from (1,0) to (-1,0). 3 d) F dr where F ( y 3y and is the line segment from (-1,0) to (,3). 3 3 e) y d ( 3y ) dy where is the path from (0,0) to (1,1) along the graph of along the graph of y. 3 y and from (1,1) to (0,0) f) F n ds where F ( z and S is y z 4 S

5 8. Green s Theorem 3 3 a) Use Green s theorem to find F dr where F ( y, 3y and is the circle with radius 3, oriented counter-clockwise (You may need the double-angle formula for cos somewhere during your computations) b) Evaluate da where is the ellipse y by using a vector field y F (, and the boundary of the ellipse. 9. Evaluate the following integrals. You can use any theorem that s appropriate: c) yzd zdy ydz where is a smooth curve from (0,0,0) to (1,4,3) d) yd dy where is the boundary of the square with vertices (0,0), (0,), (,0), and (,) e) y d ydy, where is given by r ( t) 4 cos( t),sin( t), t between 0 and Pi.

6 f) yd dy where is the boundary of the region between the graphs of y and y. 10. Prove the following: a) If F ( M (, N(, P( is any vector field where M, N, P are twice continuously differentiable then div ( curl( F)) 0 f f f b) A function (not a vector field) is called harmonic if 0. Show that for any y z 1 function the function is harmonic.