If we have \(n\) inputs and \(m\) functions such that:

\(f_i(\mathbf x)\)

The Jacobian is a matrix where:

\(J_{ij}= \dfrac{\delta f_i}{\delta x_j}\)

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

\(\mathbf F=-\nabla P\)

Scalar potentials are not unique.

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

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.

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 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.

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

Cross product of divergence with the gradient of the function.

\(\Delta f= \nabla . \nabla f\)

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

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.

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

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

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.

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.

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.

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.