Vector Field

A vector field is a vector-valued function (differentiable map) that transforms/assigns vectors to vectors.

Definition §

Vector Field ^definition

Let $X^n$ be an $n$-dimensional set. A vector field, $\vec{F}$ is a vector-valued function defined that maps $n$-dimensional vectors to $n$-dimensional vectors.

$$\vec{F}:X^n\to X^n$$

$$F(x_1,\dots ,x_n)=(f_1(x_1,\dots ,x_n),\dots ,f_2(x_1,\dots ,x_n))$$

where $f_i$ are scalar functions: $f_i:X^n\to X$ Vector fields are different from:

• Vector-valued functions: A vector function can be any functions satisfying $\vec{G}:Y^c\to X^n$. $Y$ does not have to be of dimension $n$ like for the vector field $F$.
• Scalar functions: A scalar function can take in vectors, but must always output a scalar. It is of form: $f:X^n\to Z$ where $Z$ must be of dimension 1.

• In most cases, vector fields are both continuous (all composing functions $f_i$ are continuous) and real (all $f_i$ are real scalar functions).
• The smoothness of a vector field $\vec{F}$ is given by its least smooth component function $f_i$:

Geometry §

A vector field can intuitively be seen by drawing many vectors, each starting at some point $\vec{x}\in X^n$ and being displaced by $\vec{F}(\vec{x})$.

Unlike a curve, a vector field is best visualised over some discrete points of the domain. %%🖋 Edit in Excalidraw, and the dark exported image%%

Flow Lines §

See Flow Line

Integrals Over Vector Fields §

While it isn’t possible to directly compute standard integrals (including double and triple integrals), it is possible to take the integral of a vector field $\vec{F}$ by measuring its ‘influence’ over another vector field. This is done via the dot product, which results in a scalar function which can then be integrated normally.

In the case of Path Integral, $\vec{F}$ is dotted with the tangent vector field $\vec{T}$ of the curve $C$. The more $\vec{F}$ is ‘aligned’ with $C$ (the more it is parallel to $\hat{T}$), the stronger its ‘influence’ and the higher the value of the path integral. Hence in the path integral:

$$\int \vec{F}\cdot \text{d}\vec{c}$$

The vector field is dotted against an infinitesimal vector of $C$

This process can then be extended to flux integrals, where $\vec{F}$ is now dotted with the normal vector field $\vec{N}$ of the surface, $S$. This time, the more $\vec{F}$ ‘cuts’ into the surface (is perpendicular to the surface, and thus parallel to $\vec{N}$), the higher the value of the surface integral.

Variants §

Incompressible §

An incompressible vector field $\vec{F}$ (also called solenoidal vector field) is one that has the property:

$$\nabla \cdot \vec{F}=\text{div }\vec{F}=0$$

for all points $\vec{x}$ in its domain.

Irrotational §

An irrotational vector field is defined by having:

$$\nabla \times \vec{F}=\text{curl }\vec{F}=\vec{0}$$

Conservative §

See Conservative Vector Field.

Note: Every conservative vector field is also irrotational.

Examples §

1: Vector Field away from Y-axis

Sketch the vector field:

$$\vec{F}(x,y)=(x,0)$$

Visuals

2: Vector Field towards X-axis

Sketch the vector field:

$$\vec{F}(x,y)=(0,-y)$$

Visuals

3: Composite Vector Field

Sketch the vector field:

$$\vec{F}(x,y)=(x,-y)$$

Visuals

3: Spiral Vector Field

Sketch the vector field:

$$\vec{F}(x,y)=(-y,x)$$

Visuals

#todo gravitational vector field