Find the work done by the gradient vector field F(x, y, z) = V (e² + y² + z²) in %3| mov ing a particle along the circular helix r(0) = (cos 0, sin 0,0) for 0 < 0 < 2ñ.

A solid material has thermal conductivity K in kilowatts per meter-kelvin and temperature given at each point by w(x, y, z) = 25 − 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 Vw ds. 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.) x J[, VW S -K Incorrect VwdS= 12800 kW

Find the conservative vector field for the potential function by finding its gradient. h(x, y, z) = 8x arcsin(yz)

Calculate the line integral of F(x, y) = -4i + 5j along the line segment from (-3,-4) to (-5,0). HINT: If possible, find a scalar potential. [.F. F. ds =