Consider the vector field F = yi - xj - 2k and the surface S defined to be the top half (z > 0) of the sphere x² + y² + z² = 4, with unit normal pointing down. The boundary of this surface is x² + y² = 4 which can be parametrized as a = 2 cos(t), y = -2 sin(t) for - te [0, 2π) which is traversed clockwise. Then SfsVX F. ds = AT The integer A is [hint-use Stokes Theorem] Answer: Consider the heat equation in a cylinder of radius R and height R. The end z = 0 is kept L. It is insulated on its side at p at temperature 0 and the end z = z= L is insulated. What is the appropriate boundary condition for the temperature Tat z = L? O a. T(R, 0, z, t) = 0 O b. 8T/Op=0 О с. OT = 0 dz Od. T(p, 0, 0, t) = 0. Consider the ODE F'(x) = cF(x) Find F(x) O a. Aeve + Be=√x O b. Ae O c. Ax+B Od. A cos(√cx) + B sin(√cx)
