Question

Given the scalar function Txyz) (e.g. the temperature at any point in the room), which of the three operations (div, grad, or curl) can be sensibly operated on T? For each operation that can be applied:_______. a) give a formula for the result b) explain in words how you would interpret the result. c) is the result a vector or scalar?

Answer 1

Explanation:

We have t(x,y,z) as a scalar function. Divergence and curl are operations that can be done on a vector function. So they are not necessary for a scalar function. Grad is what is used for a scalar function.

Therefore the gradient is the sensible operator on temperature compared to others.

a. Grad T = δT/ δx.(xˆ ) + δT/ δy.(yˆ) + δT/ δz.(zˆ)

b. The gradient of T(x,y,z) gives a measure of the direction and the rate at which temperature changes in all directions.

C. The result is going to be a vector quantity.

I have written out the answer to a in an attachment.