Let S be the level surface g(x, y, z) = 4 where g(x, y, z) = x4 + y® + 32º. Let F be the vector field Vg ||V9||' F = and let G be the vector field G = Vg × (0,0,3). (a) Show that the flux of F through S is equal to the surface area of S (up to a sign). (b) Compute the flux of G through S. Hint: Your answers should not be long!

Please make a graph of the gradient vector field

2. For the vector field F (x, y, z) = (x²y, xyz, −x²y²) F(x, y, z)=(x2y,xyz,-x2 y 2 ), find: (a) div (F)div (F) (b) curl (F)curl(F) (c) div (curl (F))

. Calculate the outward áux of the vector Öeld F(x; y) = 2e^xyi + 3y^3jacross the square bounded by the lines x = +-1; y =+-1 K=3

Let F(x, y) = ⟨xy^2+2x+1, x^2y − 3y⟩ be a vector field. Let C be the curve thatconsists of two line segments C1 and C2, where C1 has vertices (−1, 0) and (0, 1)and C2 has vertices (0, 1) and (1, 0). C is oriented clockwise.(a) Sketch C accurately.(b) Find the work done by F by moving a particle along the path C in the givendirection.

We are given a vector field and a parametric curve

A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at each point by w(x, y, z) = 15 - 4(x² + y² + z²) °C. Use the fact that heat flow is given by the vector field F = -KVw and the rate of heat flow across a surface S within the solid is given by -K , VwdS. Find the rate of heat flow out of a sphere of radius 1 (centered at the origin) inside a large cube of copper (K = 400 kW/(m · K)). (Use symbolic notation and fractions where needed.) -K Incorrect D VwdS= 4.104 kW

Let f be a scalar-valued function and F a vector field defined on R^3. ∇f iis the gradient of the function f. Show whether the operations shown produce a scalar function, a vector field, or absolute nonsense.

Consider the vector field v(x)=(y/(x^2+y^2))i − (x/(x^2+y^2)) j + 4k. Calculate ∫Cv dr, where C is the straight line from (1,0,1)to (0,1,2).

Find all points (a, b, c) on the sphere x2 + y2 + z2 = R2 where the vector field F = yz2i + xz2j + 2xyzk is normal to the surface and F(a, b, c) ≠ 0.