Student Solution Manual for Mathematical Methods for Physics and Engineering, Third Edition

5

Partial differentiation

5.1  Using the appropriate properties of ordinary derivatives, perform the following.

(a)  Find all the first partial derivatives of the following functions f(x, y):

(i) x²y, (ii) x² + y² + 4, (iii) sin(x/y), (iv) tan^–1(y/x),

(v) r(x, y, z) = (x² + y² + z²)^1/2.

(b)  For (i), (ii) and (v), find ∂²f/∂x², ∂²f/∂y² and ∂²f/∂x∂y.

(c)  For (iv) verify that ∂²f/∂x∂y = ∂²f/∂y∂x.

These are all straightforward applications of the ...