Vector Calculus Key concepts

Definition

Vector calculus is calculus for fields: quantities spread across space.

Normal calculus asks how a function changes along one variable.

Vector calculus asks how quantities change across space: left/right, up/down, in/out.

Vectors

A vector has size and direction. In ${\mathbb{R}}^3$:

$$\mathbf{v}=(v_x,v_y,v_z)$$

The dot product tells you how aligned two vectors are:

$$\mathbf{a}\cdot \mathbf{b}=\Vert \mathbf{a}\Vert \Vert \mathbf{b}\Vert \cos \theta$$

The cross product gives a new vector perpendicular to both input vectors. Its size is the area of the parallelogram made by them.

Scalar and vector fields

A scalar field puts one number at every point.

Example: temperature

$$T(x,y,z)$$

A vector field puts one vector at every point.

Examples:

• Velocity in a fluid
• electric field in Electromagnetism index
• force field around a mass or charge

Gradient, divergence, curl

The Del operator ∇ packages the main spatial derivatives.

• gradient: where a scalar field climbs fastest
• divergence: how much a vector field spreads out or sinks in
• curl: how much a vector field rotates locally

Tiny mental model:

• gradient: hill slope
• divergence: source or drain
• curl: swirl

Flux and circulation

Flux measures how much field passes through a surface.

Circulation measures how much field pushes around a loop.

If flux is “how much goes through the net”, circulation is “how much goes round the track”.

Conservative fields

A vector field is conservative if it comes from a potential:

$$\mathbf{F}=\nabla \varphi$$

or, in physics,

$$\mathbf{F}=-\nabla U$$

depending on convention.

The important bit: if a field is conservative, path does not matter. Only the start and end points matter.

That is the geometric idea behind Potential energy.

Flat and curved spaces

Most of this assumes Euclidean spaces: normal flat space with nice Cartesian coordinates.

On Manifolds, derivatives need extra machinery such as a metric and connection.