# 17. Divergence, Curl and Potentials

## e. Generalizing Antiderivatives

### Antiderivatives

For a function of \(1\) variable, \(F(x)\), if we take the derivative, we get another function \(f(x)=\dfrac{dF}{dx}\). Conversely:

An antiderivative of a function \(f(x)\) is any function \(F(x)\) whose derivative is \(f(x)\), i.e. \[ \dfrac{dF}{dx}=f(x) \]

If \(f(x)\) is continuous on some interval, then an antiderivative is guaranteed to exist in that interval. An antiderivative is never unique, but any two antiderivatives will always differ by a constant: \[ \text{If} \qquad \dfrac{dF}{dx}=\dfrac{dG}{dx}=f(x) \qquad \text{then} \qquad F(x)=G(x)+C \]

### Scalar Potentials

For a function of several variables (i.e. a scalar field), \(f(x,y,z)\), if we take all the partial derivatives, we get the gradient: \[ \vec F=\vec\nabla f =\left\langle \partial_x f,\partial_y f,\partial_z f\right\rangle \] which is a vector field. Conversely:

A scalar potential for a vector field \(\vec F(x,y,z)\) is any function \(f(x,y,z)\) whose gradient is \(\vec F\), i.e. \[ \vec\nabla f=\vec F \]

When finding antiderivatives, it is customary to take the original function as lower case \(f\) and the antiderivative as upper case \(F\).

When finding scalar potentials, it is customary to take the original vector field as upper case \(\vec F\) and the scalar potential (analogous to the antiderivative) as lower case \(f\).

This reversal of upper and lower case is just something one gets used to.

Unlike antiderivatives, even if the vector field \(\vec F\) has continuous derivatives of all orders, there is no guarantee that it will have a scalar potential. Like antiderivatives, if there is a scalar potential, it will not be unique and again the difference between two scalar potentials will be a constant. On the next two pages, we will discuss the existence and non-uniqueness of scalar potentials and how to find them.

### Vector Potentials

In discussing several variables, we can also take a derivative of a vector field \(\vec A(x,y,z)\). It is most interesting to take this derivative to be the curl: \[ \vec F=\vec\nabla\times\vec A \] which is also a vector field. Conversely:

A vector potential for a vector field \(\vec F(x,y,z)\) is any other vector field \(\vec A(x,y,z)\) whose curl is \(\vec F\), i.e. \[ \vec\nabla\times\vec A=\vec F \]

Like scalar potentials, even if the vector field \(\vec F\) has continuous derivatives of all orders, there is no guarantee that it will have a vector potential. Further, if there is a vector potential, it will not be unique but this time the difference between two vector potentials will not be just a constant. On the subsequent two pages, we will discuss the existence and non-uniqueness of vector potentials and how to find them.

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