Verify the Divergence Theorem for the vector field F = (-2, 3, z³) on the sphere x² + y² +2²2² = 4 with outward pointing normal vectors. Note, you may use a change of variables on the triple integral side.

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 , 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/(mK)). (Use symbolic notation and fractions where needed.) -K Incorrect 1₁² VwdS= 12800T 32 Sille Jour me que an kW

Compute the flux of the vector field F = 5yi +37-5xzk through the surface S, which is the surface y = x² + 2², with x² + 2² ≤ 1, oriented in the positive y-direction. flux = You can get a hint after 2 incorrect answers.

Show that Green's Theorem holds for the vector fieldF(x, y)=(x +y, 3x) at a square of side 2.