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Multivariate Calculus // Fields

VECTOR
CALCULUS

Moving away from static surfaces, Vector Calculus deals with environments where every point in space has both a magnitude and a direction. It is the absolute mathematical foundation of fluid dynamics and electromagnetism.

Divergence (Flux Density)

Imagine a vector field as a flowing body of water. Divergence measures how much a fluid is expanding from or compressing into a specific point. It asks the question: "Is this point a source, or a sink?"

div F=∇⋅F=∂P∂x+∂Q∂y\text{div } \mathbf{F} = \nabla \cdot \mathbf{F} = \frac{\partial P}{\partial x} + \frac{\partial Q}{\partial y}

If you drop a mathematical sensor into a field with positive divergence (a Source), the sensor will physically expand outward as the flow pushes away from the center.

Curl (Circulation Density)

While Divergence measures expansion, Curl measures macroscopic rotation. It calculates the tendency of the vector field to swirl around a given point.

curl F=∇×F=(∂Q∂x−∂P∂y)k\text{curl } \mathbf{F} = \nabla \times \mathbf{F} = \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) \mathbf{k}
If you drop a "paddlewheel" into a field with non-zero curl (a Vortex), the uneven flow of the field will cause the paddlewheel to physically spin on its axis.

Divergence & Curl

Vector Field Analysis

F(x,y) = ⟨-y, x⟩
div F = 0 (No expansion)
curl F = 2 (Spins!)

The Fundamental Theorems

The crowning achievements of vector calculus. These theorems relate the boundary of an object to its interior.

Green's Theorem

Relates a macroscopic line integral around a closed curve to a double integral over the plane region it encloses.

Stokes' Theorem

The 3D generalization of Green’s Theorem. Relates the surface integral of the curl to the line integral around the boundary.

Divergence Theorem

Also known as Gauss’s Theorem. Relates the flux of a vector field through a closed surface to the divergence within the enclosed volume.