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. \[ \vec\nabla f=\vec F \] Note on upper and lower case.

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.