Question

Let z = g(u,v) and u(r,s),v(r,s). How many terms are there in the expression for ∂z/∂r?

a) 1
b) 2
c) 3
d) 4

Answer 1

The number of terms in the expression for ∂z/∂r is 1.

Step-by-step explanation:

In order to find the number of terms in the expression for ∂z/∂r, we need to consider the chain rule for partial derivatives. The chain rule states that if z = g(u,v) and u = u(r,s), v = v(r,s), then ∂z/∂r = (∂z/∂u)(∂u/∂r) + (∂z/∂v)(∂v/∂r).

Since z = g(u,v), there is only one term for ∂z/∂r because it does not contain any u or v. Therefore, the answer is option a) 1.

The question concerns the application of the chain rule in partial differentiation within the context of multivariable calculus. When given that z is a function g(u,v), and both u and v are functions of r and s, the expression for ∂z/∂r involves the partial derivatives of z with respect to u and v, as well as the partial derivatives of u and v with respect to r. Using the chain rule, the derivative of z with respect to r is ∂z/∂r = (∂z/∂u)(∂u/∂r) + (∂z/∂v)(∂v/∂r), which consists of two terms.