Finding the gradient of a scalar field in cylindrical coordinates

Question

How do I find the gradient of the following scalar field in cylindrical polar coordinates?

$\ f(x,y,z)=2z-3x^2-4xy+3y^2$

Should I express it in polar form first, then take the partial derivatives? Is there a quicker way?

Answer 1

You can use chain rule to get:

$$f_r = \frac{\partial f}{\partial x} \cdot \frac{\partial x}{\partial r} + \frac{\partial f}{\partial y} \cdot \frac{\partial y}{\partial r} + \frac{\partial f}{\partial z} \cdot \frac{\partial z}{\partial r}$$

Similarly:

$$f_\theta = \frac{\partial f}{\partial x} \cdot \frac{\partial x}{\partial \theta} + \frac{\partial f}{\partial y} \cdot \frac{\partial y}{\partial \theta} + \frac{\partial f}{\partial z} \cdot \frac{\partial z}{\partial \theta}$$

$$f_z = \frac{\partial f}{\partial x} \cdot \frac{\partial x}{\partial z} + \frac{\partial f}{\partial y} \cdot \frac{\partial y}{\partial z} + \frac{\partial f}{\partial z} \cdot \frac{\partial z}{\partial z}$$