Vector Calculus index

Vector calculus is calculus for fields.

Instead of asking “how does this function change as $x$ changes?”, it asks:

• where does this field climb?
• where does it spread out?
• where does it rotate?
• how much flows through this surface?
• how much pushes around this loop?

This is the language behind fluids, Electromagnetism index, gravity, heat, and a lot of optimisation in ML.

Core route

• Vector Calculus Key concepts - main ideas first
• Del operator ∇ - compact notation for spatial derivatives
• Curvilinear coordinate systems - Cartesian, cylindrical, and spherical coordinates
• Vector Calculus Equations and definitions - formula sheet
• Vector Calculus Examples - small worked examples
• Vector Calculus Common pitfalls - sign, coordinate, and interpretation traps

Prerequisites

Useful background:

• Euclidean spaces
• Line
• Plane
• Space (3D)
• Trigonometry
• Derivative
• Integral

Stats Equations and definitions and ML Equations and definitions use gradients when fitting models.

Physics bridges

• Velocity is the derivative of position and can be represented as a vector field.
• Lorentz force uses the vector cross product $\mathbf{v}\times \mathbf{B}$ to set magnetic-force direction.
• Central force problem uses radial vectors, gradients of potentials, and planar orbital geometry.
• Electromagnetism index uses divergence and curl in Maxwell-style laws.
• Solenoidal magnetic fields and no magnetic monopoles uses $\nabla \cdot \mathbf{B}=0$.
• Residuals in NF2 and PINNs uses differential residuals as loss terms.

Core intuition

Derivatives of fields tell you whether quantities climb, spread out, rotate, or smooth out.

That is basically the whole game.