# Gradient

## Table of Contents

The gradient represents the direction and magnitude of the steepest ascent of the function \( \small f \) at any given point.

## Cartesian Coordinates #

The gradient of a scalar function \( \small f \) in cartesian coordinates is given by:

$$ \nabla f = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k} $$

Where:

- \( \small \nabla \) represents the del operator (also known as the gradient operator),
- \( \small \mathbf{i}, \mathbf{j}, \mathbf{k} \) are the unit vectors in the x, y, and z directions respectively, and
- \( \partial \) represents the partial derivative operator.

## Cylindrical Coordinates #

The gradient of a scalar function \( \small f \) in cylindrical coordinates is given by:

$$ \nabla f = \frac{\partial f}{\partial \rho} \hat{\rho} + \frac{1}{\rho} \frac{\partial f}{\partial \varphi} \hat{\varphi} + \frac{\partial f}{\partial z} \hat{z}$$

Where:

- \( \small \rho \) is the radial coordinate, \( \small \varphi \) is the azimuthal coordinate, and \( \small z \) is the vertical coordinate,
- \( \hat{\rho}, \hat{\varphi}, \hat{z} \) are the unit vectors in the \( \small \rho, \varphi, z \) directions respectively.

## Spherical Coordinates #

The gradient of a scalar function \( \small f \) in spherical coordinates is given by:

$$ \nabla f = \frac{\partial f}{\partial r} \hat{r} + \frac{1}{r} \frac{\partial f}{\partial \theta} \hat{\theta} + \frac{1}{r \sin \theta} \frac{\partial f}{\partial \varphi} \hat{\varphi} $$

Where:

- \( \small r \) is the radial coordinate, \( \small \theta \) is the polar coordinate, and \( \small \varphi \) is the azimuthal coordinate, and
- \( \hat{r}, \hat{\theta}, \hat{\varphi} \) are the unit vectors in the \( \small r, \theta, \varphi \) directions respectively.