Question

# 1. Consider a cubic room of side length a, the vertices of which are (0,0,0), (a,0,0), (0, a, 0), (0, 0, a),(a, a, 0), (a, 0, a), (0, a, a), and (a, a, a). At a certain moment, the temperature within this room is given by the scalar field T(x, y, z)= Toxy z, where To is a constant. (a) What are the physical dimensions of the constant To? (b) Where is the hottest point in the room? (c) Find the functions Tr-o(y, z), Tz-a (y, z), Ty=0(x, 2), Ty-a (x, z), T₂-0(x, y), Tz-a (x, y), which give the temperature profile on each of the six walls of the room. (d) In the centre of the room (i.e. the body-centre of the cube), what is the direction of maximum increase of temperature? Comment on your answer./n2. Find the curl of the vector field A = x (xê, +yê, + zê₂). 3. Consider the vector field F = yêr+zêy + xêz. (a) Calculate the divergence and the curl of this vector field. Comment on your answers. (b) Evaluate the line integral fr.a, where the contour C consists of: a straight line from the origin to (0, 0, b); a quarter-circle centred at the origin from (0, 0, b) to (0, 6, 0); and a straight line from (0, 6, 0) to the origin, i. explicitly; ii. using Stokes' Theorem. Check that your answers agree.   /n4. Use the divergence theorem to evaluate the surface integral ff (3x ê₂ + 2y êy) · ďªs, where S is the surface of a sphere of radius 36 centred on the origin. 5. Find the divergence of the vector field A = x (zê₂ + xêy + yêz).  Fig: 1  Fig: 2  Fig: 3