Let v = (x³ cos z, 4 - 3x²y cos z − 3yz² sin x, z³ sin x) be the velocity field of a fluid. Compute the flux of v across the surface x² + y + z² = 9 where y > 0 and the surface is oriented away from the origin. Note that this surface is not closed.

+25y²,0, 10- through the surface 3 Find the outward flux of F = =1.

Suppose F(x, y) =-yi+aj and C is the line segment from point P = (3,0) to Q = (0,4). (a) Find a vector parametric equation r(t) for the line segment C so that points P and Q correspond to t= 0 and t = 1, respectively. F(t) = (b) Using the parametrization in part (a), the line integral of F along Cis So - [* F(F(t)) - F' (t) dt = SC dt with limits of integration a = and b= F. . dr = (c) Evaluate the line integral in part (b). (d) What is the line integral of F around the clockwise-oriented triangle with corners at the origin, P, and Q? Hint: Sketch the vector field and the triangle.

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