1. Find the gradient of p(x, y, z) = 2xy + ze"; evaluate the gradient at...
Test the divergence theorem for the function as your volume the cube as shown. v = (xy)x + (2yz)y +3x), Take 3. Compute the line integral of v=6x + yz2j) + (3), + z)2 Along the triangular path shown
5. Let F (y”, 2xy + €35, 3yes-). Find the curl V F. Is the vector field F conservative? If so, find a potential function, and use the Fundamental Theorem of Line Integrals (FTLI) to evaluate the vector line integral ScF. dr along any path from (0,0,0) to (1,1,1). 6. Compute the Curl x F = Q. - P, of the vector field F = (x4, xy), and use Green's theorem to evaluate the circulation (flow, work) $ex* dx +...
2.) Show that the fundamental theorem of divergences (aka Gauss's theorem, aka Green's theorem), shown below, holds for the (vector) function v from the previous problem. (Use the cube shown below as the basis for your work; the cube has sides of length 3.) fundamental theorem of divergences (V.v)dr v-da 24 A(v) (ii) 47 (iv) (ii) (vi) 1.) Calculate the divergence of the following (vector) function: v (xy)x +(2yz)y+ (3xz)z (NOTE: x, y, and z are Cartesian unit vectors.) 2.)...
Consider the following function 6 f(x, y,z)=z - x? cos(my) + xy? (i) Find the gradient of the function f(x, y, z) at the point P,(2,-1,-7). (ii) Find the directional derivative of f(x, y, z) at P,(2,-1,-7) along the direction of the vector ū = 2î+j+2k. (iii) Find the equation of the tangent plane to the surface given below at the point P,(2,-1, -7). 6 :- xcos(ty) + = 0 xy
Consider the following. f(x, y, z) = xe3yz, P(2,0,1), u } (a) Find the gradient of f. Vf(x, y, z) = (b) Evaluate the gradient at the point P. VF(2,0, 1) = (c) Find the rate of change of fat P in the direction of the vector u. Duf(2, 0, 1) =
Let F(x, y, z)=(x + 2xy, y – 3zy, z + x2) Find the divergence and curl of F.
a) Find the gradient of f(x, y, z) = 4x + 8y + 3z – 24 and indicate it at point P = (0,3,0) Draw the function in 3D, draw the plane that is generated when f(x,y,z)=0, start with the lines on the xy, yz, and xy planes. X у
a. Sketch the solid S:= {[x; y; z] in |R3 | x,y,z ≥ 0, and 2x + 4 y + 2z ≤ 12}. b. Using your calculator evaluate, i) as a triple integral and ii) by the divergence theorem, the volume of S. c. Find i)the surface area of the solid S and ii)the flux thru the top of S due to the vector field F, where F(x,y,z) = ( x + yz , y + xz , z +...
Problem 5. Let F(r,y) (e-v-v sinzy) ?-(ze-s + z sin zyj (1) Show that F is a gradient field. (2) Find a potential function f for it (3) Use the potential function f to evaluate F-ds, where x is the path x(t) = (t,t2) for 0sts1. (NO credit for any other method.)
5 Use the Divergence theorem to find the outward flux. a. F(a, y,z)-(6x2+ + 2xy, 2y + xz, 4x2y); G: The solid cut from the first octant by the cylinder x2+y - 4 and the plane 3. (In(x2+Уг),-2z arctan(y/x), z (x2 +y2); G:The solid between the b. F(r, y, z) Vx + y*); G: The solid between the cylinders x2 + y.2 1 and x2+ y2 2, -1szs4. c Fxy)-(2xy', 2x'y, -): G: The solid bounded by the cylinder x?1...