MATH 2110Q. Multivariable Calculus. Fall 2009 Professor Leibowitz
16.5 Vector Differentiation Operators Homework Solutions and comments
In addition to illustrating the definitions and properties of the curl and divergence operators, the
homework problems give some practice in finding partial derivatives. In Cartesian coordinates, curl F =
F and div F = F where is symbolically x i + y j + zk ( x means , etc). So if F = P i + Q j +
R k then curl F = (Ry โ Qz) i + (Pz Rx ) j + (Qx Py) k and div F = Px + Qy + Rz .
#1. F(x , y, z) = xyz i โ x2y k. Notice that the coefficient of j is 0.
Curl F = โ x2 i +(2xy + xy) j + (โ xz) k = <โ x2 , 3 xy, โ xz> and div F = yz + 0 + 0 = yz.
#6. F(x , y, z) = exysin z j + y arctan(x/z) k.
The steps are the same as in #1 but the calculus facts that are used are more intricate. In particular we
need to know the derivative of eu where u is a function. By the chain rule, the derivative of eu is eu times
the derivative of u. Also, the derivative of is (1/(1 + u2)) times the derivative of u.
#15. F(x , y, z) = 2xy i + (x2 + 2yz) j + y2 k. We find that Curl F = <2y โ 2y, 0 โ 0, 2x โ 2x> = 0.
To find f so that grad = F, set up fx = 2xy , fy = x2 + 2yz, and fz = y2. Anti-differentiate each relation,
treating all the other variables as though they were constants. You find that f = x2y + 1(y,z) , f = x2y +
y2z + 2(x , z) , f = y2z + 3(x , y) for some functions as indicated. Compare the three equal expressions
an d you see that f(x, y, z) = x2y + y2z + a constant.
#20. G1 = yz i + xyz j + xy k. Since the divergence of G1 is xz instead of 0,but div(curl G) always is 0, the
answer is No; G1 is not the curl of a vector field.