Gradient

A scalar field is a function that assigns a number to every position in space. Consider a scalar field $\phi(x,y,z)$. The gradient of this scalar field is defined as,

$$ \nabla \phi = \frac{\partial \phi }{\partial x}\hat{x}+\frac{\partial \phi }{\partial y}\hat{y}+\frac{\partial \phi }{\partial z}\hat{z}.$$

The gradient of a scalar field is a vector field. If the scalar field is an topological map that shows the altitude at every point, the gradient at every point is in the direction that goes up the steepest. If the scalar field is the pressure, minus the gradient points in the direction that the wind blows. If the scalar field is the temperature, minus the gradient points in the direction that the heat flows. If the scalar field is the potential energy, minus the gradient is the force.

$\phi(x,y,z)=$  
$\nabla \phi = $ () $\hat{x}$ + () $\hat{y}$ + () $\hat{z}$

Example 1

The gravitational potential energy is,

$$E_{pot} =- \frac{Gm_1m_2}{r},$$ $$E_{pot} =- \frac{Gm_1m_2}{\sqrt{x^2+y^2+z^2}},$$ $$-\nabla E_{pot} = -\frac{Gm_1m_2}{\left( x^2+y^2+z^2\right)^{3/2}}\left(x\hat{x}+y\hat{y}+z\hat{z}\right).$$

The force is,

$$-\nabla E_{pot} = \vec{F} = -\frac{Gm_1m_2}{r^2}\hat{r}.$$

Example 2

The Coulomb potential energy is,

$$E_{pot} = \frac{q_1q_2}{4\pi \epsilon_0 r},$$ $$E_{pot} =\frac{q_1q_2}{4\pi \epsilon_0\sqrt{x^2+y^2+z^2}},$$ $$-\nabla E_{pot} = \frac{q_1q_2}{4\pi \epsilon_0\left( x^2+y^2+z^2\right)^{3/2}}\left(x\hat{x}+y\hat{y}+z\hat{z}\right).$$

The force is,

$$-\nabla E_{pot} = \vec{F} = \frac{q_1q_2}{4\pi \epsilon_0r^2}\hat{r}.$$

Example 3

The potential energy stored in a linear spring is,

$$E_{pot} = \frac{kx^2}{2}.$$

The force is,

$$-\nabla E_{pot} = \vec{F} = -kx.$$
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