1) Given the following potential function, =x- x + y +2 a. Determine and state the...

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1) Given the following potential function, ø=x- √x² + y² +2²² a. Determine and state the associated conservative vector field, F =V6. b. Determine and state the curl of this vector field. Explain what the results of these calculations imply. 2) Consider the vector field F = (6x² - 6xy, 4 - 3x²) a. Determine and state the divergence of this vector field. b. Show that this vector field is conservative. c. Find the potential function associated with this conservative vector field. 3) Consider the vector field F(x, y, z)=(2xy, x² +2²,2yz) a. Show that this vector field is conservative. b. Find the potential function associated with this conservative vector field. c. Calculate the circulation of this vector field around the curve given by the intersection of the cone = 7/4 and the sphere p=1. You may use the results of part a. or b. and integration theorems if it helps. 4) Write down Green's Thm. Explain when Green's Thm applies and when it does not. Give an example of Green's Thm showing the function, the integral and drawing the region. 5) Determine and state the work done by the vector field over the given path in each situation. A) F = (3x-y, 2x). Move from (0,0) to (3,6) then from (3,6) to (9,0). B) F = (6x² - 6xy, 4-3x2) Move from (0,0) to (4,4) then from (4,4) to (6,2). A 0 -2 C) F = (3y,4x) Move from (0,0) to (6,3) then from (6,3) to (0,6) then from (0,6) back to (0,0). 2 6 0 1) Given the following potential function, ø=x- √x² + y² +2²² a. Determine and state the associated conservative vector field, F =V6. b. Determine and state the curl of this vector field. Explain what the results of these calculations imply. 2) Consider the vector field F = (6x² - 6xy, 4 - 3x²) a. Determine and state the divergence of this vector field. b. Show that this vector field is conservative. c. Find the potential function associated with this conservative vector field. 3) Consider the vector field F(x, y, z)=(2xy, x² +2²,2yz) a. Show that this vector field is conservative. b. Find the potential function associated with this conservative vector field. c. Calculate the circulation of this vector field around the curve given by the intersection of the cone = 7/4 and the sphere p=1. You may use the results of part a. or b. and integration theorems if it helps. 4) Write down Green's Thm. Explain when Green's Thm applies and when it does not. Give an example of Green's Thm showing the function, the integral and drawing the region. 5) Determine and state the work done by the vector field over the given path in each situation. A) F = (3x-y, 2x). Move from (0,0) to (3,6) then from (3,6) to (9,0). B) F = (6x² - 6xy, 4-3x2) Move from (0,0) to (4,4) then from (4,4) to (6,2). A 0 -2 C) F = (3y,4x) Move from (0,0) to (6,3) then from (6,3) to (0,6) then from (0,6) back to (0,0). 2 6 0

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1 a To determine the conservative vector field we need to find a vector field F F1 F2 such that F 0 Given the potential function x x 1 we can find the ... View the full answer