# Multivariate differentiation of vector fields

## Partial differentiation of vector fields

### Jacobian matrix

If we have $$n$$ inputs and $$m$$ functions such that:

$$f_i(\mathbf x)$$

The Jacobian is a matrix where:

$$J_{ij}= \dfrac{\delta f_i}{\delta x_j}$$

## Scalar potential

### Scalar potential

Given a vector field $$\mathbf F$$ we may be able to identify a scalar field $$P$$ such that:

$$\mathbf F=-\nabla P$$

### Non-uniqueness of scalar potentials

Scalar potentials are not unique.

If $$P$$ is a scalar potential of $$\mathbf F$$, then so is $$P+c$$, where $$c$$ is a constant.

### Conservative vector fields

Not all vector fields have scalar potentials. Those that do are conservative.

For example if a vector field is the gradient of a scalar height function, then the height is a scalar potential.

If a vector field is the rotation of water, there will not be a scalar potential.

## Divergence

### Divergence

This takes a vector field and produces a scalar field.

It is the dot product of the vector field with the del operator.

$$div F = \nabla . F$$

Where $$\nabla =(\sum_{i=1}^n e_i\dfrac{\delta }{\delta x_i})$$

$$div F = \sum_{i=1}^n e_i\dfrac{\delta F_i}{\delta x_i}$$

### Divergence as net flow

Divergence can be thought of as the net flow into a point.

For example, if we have a body of water, and a vector field as the velocity at any given point, then the divergence is $$0$$ at all points.

This is because water is incompressible, and so there can be no net flows.

Areas which flow out are sources, while areas that flow inwards are sinks.

### Solenoidal vector fields

If there is no divergence, then the vector field is called solenoidal.

### The Laplace operator

Cross product of divergence with the gradient of the function.

$$\Delta f= \nabla . \nabla f$$

$$\Delta f= \sum_{i=1}^n \dfrac{\delta^2 f}{\delta x^2_i}$$

## Curl

### Curl

The curl of a vector field is defined as:

$$curl \mathbf F=\nabla \times \mathbf F$$

Where: $$\nabla =(\sum_{i=1}^n e_i\dfrac{\delta }{\delta x_i})$$

And: $$\mathbf x\times \mathbf y=\||\mathbf x|| ||\mathbf y|| \sin(\theta )\mathbf n$$

The curl of a vector field is another vector field.

The curl measures the rotation about a given point. For example if a vector field is the gradient of a height map, the curl is $$0$$ at all points, however for a rotating body of water the curl reflects the rotation at a given point.

### Divergence of the curl

If we have a vector field $$\mathbf F$$, the divergence of its curl is $$0$$:

$$\nabla . (\nabla \times \mathbf F)=0$$

### Vector potential

Given a vector field $$\mathbf F$$ we may be able to identify another vector field $$A$$ such that:

$$\mathbf F =\nabla \times \mathbf A$$

Existence:

We know that the divergence of the curl for any vector field is $$0$$, so this applies to $$A$$:

$$\nabla . (\nabla \times \mathbf A)=0$$

Therefore:

$$\nabla . \mathbf F= 0$$

This means that if there is a vector potential of $$\mathbf F$$, then $$\mathbf F$$ has no divergence.

### Non-uniqueness of vector potentials

Vector potentials are not unique.

If $$\mathbf A$$ is a vector potential of $$\mathbf F$$, then so is $$\mathbf A + \nabla c$$, where $$c$$ is a scalar field and $$\nabla c$$ is its gradient.

### Conservative vector fields

Not all vector fields have scalar potentials. Those that do are conservative.

For example if a vector field is the gradient of a scalar height function, then the height is a scalar potential.

If a vector field is the rotation of water, there will not be a scalar potential.

### Hodge stars

The Hodge star operator is a generalisation of cross product. In 3d space if we have a plane, we can get a vector perpendicular and visa versa. Generally, we are in $$n$$-dimensional space and we input $$k$$ vectors and get out $$n-k$$ vectors.