∞Calculus IV

Vector Calculus Identities

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Why This Matters

Vector calculus identities aren't just formulas to memorize—they're the fundamental language connecting gradient, divergence, curl, and the major integral theorems that form the backbone of Calculus IV. When you're working through problems involving fluid flow, electromagnetic fields, or conservation laws, these identities let you transform complex expressions into simpler ones and convert between different types of integrals. You're being tested on your ability to recognize which identity applies, manipulate expressions efficiently, and understand the geometric meaning behind the algebra.

The key insight is that these identities reveal deep relationships between scalar and vector fields. The gradient tells you how things change directionally, divergence measures sources and sinks, and curl captures rotation. The major theorems—Green's, Stokes', and the Divergence Theorem—connect local behavior to global behavior across boundaries. Don't just memorize the formulas; know what each identity means physically and when to deploy it on an exam.

Fundamental Differential Operators

These three operators are your building blocks. Every other identity in vector calculus is built from combinations of gradient, divergence, and curl. Master these first, and the rest follows naturally.

Gradient of a Scalar Field

$\nabla f$ points in the direction of steepest increase—its magnitude gives the rate of change in that direction
Produces a vector from a scalar—this is the only operator that "upgrades" a scalar field to a vector field
Perpendicular to level surfaces—if $f(x,y,z) = c$ defines a surface, $\nabla f$ is normal to it at every point

Divergence of a Vector Field

$\nabla \cdot \mathbf{F}$ measures net outward flux per unit volume—positive means source, negative means sink
Produces a scalar from a vector—this "downgrades" a vector field back to a scalar field
Zero divergence means incompressible flow—no fluid is created or destroyed, crucial for conservation laws

Curl of a Vector Field

$\nabla \times \mathbf{F}$ measures local rotation or circulation density—the axis of rotation points along the curl vector
Produces a vector from a vector—maintains the "vector nature" of the field
Zero curl means conservative (irrotational) field—the field can be written as a gradient of some potential function

Compare: Divergence vs. Curl—both measure local behavior of vector fields, but divergence captures expansion/compression while curl captures rotation. If an FRQ asks about fluid flow, identify whether the question involves sources/sinks (use divergence) or vortices (use curl).

Second-Order Operations

When you apply differential operators twice, you get powerful tools for analyzing how fields behave relative to their surroundings. The Laplacian is especially exam-critical.

Laplacian of a Scalar Field

$\nabla^2 f = \nabla \cdot (\nabla f)$ measures deviation from local average—positive means the point is a local minimum relative to neighbors
Appears in every major PDE—Laplace's equation, heat equation, wave equation, and Schrödinger equation all feature $\nabla^2$
In Cartesian coordinates: $\nabla^2 f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}$ —sum of second partials

Compare: Gradient vs. Laplacian—gradient gives direction and rate of change (a vector), while Laplacian tells you whether you're at a local extremum relative to surroundings (a scalar). The Laplacian is the divergence of the gradient, combining both operations.

The Big Three Integral Theorems

These theorems convert between different types of integrals and connect local differential behavior to global integral behavior. Expect at least one of these on every Calc IV exam.

Green's Theorem

Relates line integrals to double integrals in 2D— $\oint_C \mathbf{F} \cdot d\mathbf{r} = \iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA$
The 2D version of Stokes' theorem—works for flat regions in the $xy$ -plane with simple closed curves
Two forms: circulation and flux—circulation form measures rotation, flux form measures flow across the boundary

Stokes' Theorem

Connects surface integrals of curl to line integrals around the boundary— $\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r}$
Orientation matters—use the right-hand rule to match surface normal with curve direction
Generalizes Green's theorem to 3D surfaces—the surface doesn't have to be flat

Divergence Theorem

Converts volume integrals to surface integrals— $\iiint_V (\nabla \cdot \mathbf{F}) \, dV = \oiint_S \mathbf{F} \cdot d\mathbf{S}$
Also called Gauss's theorem—fundamental for deriving conservation laws in physics
The surface must be closed—think of a balloon or a box, not an open bowl

Compare: Stokes' vs. Divergence Theorem—Stokes' relates a surface integral to a line integral (boundary is a curve), while Divergence relates a volume integral to a surface integral (boundary is a closed surface). Both convert $(n)$ -dimensional integrals to $(n-1)$ -dimensional boundary integrals.

Product Rules for Vector Operations

Just like single-variable calculus has product rules, vector calculus has its own versions. These show up constantly when simplifying expressions.

Product Rule for Divergence

$\nabla \cdot (\phi \mathbf{F}) = (\nabla \phi) \cdot \mathbf{F} + \phi (\nabla \cdot \mathbf{F})$ —scalar times vector field requires both terms
Analogous to $(fg)' = f'g + fg'$ —the gradient "differentiates" the scalar, divergence "differentiates" the vector
Essential for integration by parts in vector calculus—used to derive weak forms of PDEs

Product Rule for Curl

$\nabla \times (\phi \mathbf{F}) = (\nabla \phi) \times \mathbf{F} + \phi (\nabla \times \mathbf{F})$ —note the cross product in the first term
Gradient of scalar crossed with vector field—this term vanishes if $\phi$ is constant
Key for electromagnetic derivations—appears when analyzing fields with spatially varying coefficients

Compare: Divergence product rule vs. Curl product rule—both split into two terms, but divergence uses dot products throughout while curl uses a cross product in the first term. Remember: the operator type (dot vs. cross) stays consistent.

Vector Product Identities

These identities help you manipulate expressions involving multiple vector fields. Memorize the BAC-CAB rule—it's a lifesaver.

Vector Triple Product

$\mathbf{A} \times (\mathbf{B} \times \mathbf{C}) = \mathbf{B}(\mathbf{A} \cdot \mathbf{C}) - \mathbf{C}(\mathbf{A} \cdot \mathbf{B})$ —the famous "BAC-CAB" rule
Result is a vector in the plane of $\mathbf{B}$ and $\mathbf{C}$ —perpendicular to $\mathbf{A}$
Order matters— $(\mathbf{A} \times \mathbf{B}) \times \mathbf{C}$ gives a different result

Scalar Triple Product

$\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})$ equals the volume of the parallelepiped—formed by the three vectors
Zero if vectors are coplanar—this is a quick test for linear dependence in 3D
Cyclic permutations are equal— $\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = \mathbf{B} \cdot (\mathbf{C} \times \mathbf{A}) = \mathbf{C} \cdot (\mathbf{A} \times \mathbf{B})$

Compare: Scalar vs. Vector triple product—scalar triple product gives a number (volume), vector triple product gives a vector. Both involve three vectors, but the operations and outputs are completely different.

Advanced Differential Identities

These identities involve combinations of fields and operators. They're essential for simplifying complex expressions on exams.

Gradient of Dot Product

$\nabla (\mathbf{A} \cdot \mathbf{B}) = (\mathbf{A} \cdot \nabla)\mathbf{B} + (\mathbf{B} \cdot \nabla)\mathbf{A} + \mathbf{A} \times (\nabla \times \mathbf{B}) + \mathbf{B} \times (\nabla \times \mathbf{A})$ —four terms total
Simplifies when fields are irrotational—if curls vanish, only the first two terms remain
Appears in energy and Lagrangian mechanics—connects kinetic energy gradients to forces

Divergence of Cross Product

$\nabla \cdot (\mathbf{A} \times \mathbf{B}) = \mathbf{B} \cdot (\nabla \times \mathbf{A}) - \mathbf{A} \cdot (\nabla \times \mathbf{B})$ —curls appear, not divergences
Zero if both fields are irrotational—conservative fields have zero curl
Key identity in electromagnetic theory—used in Poynting vector derivations

Curl of Cross Product

$\nabla \times (\mathbf{A} \times \mathbf{B}) = (\mathbf{B} \cdot \nabla)\mathbf{A} - (\mathbf{A} \cdot \nabla)\mathbf{B} + \mathbf{A}(\nabla \cdot \mathbf{B}) - \mathbf{B}(\nabla \cdot \mathbf{A})$ —four terms
Involves both divergences and directional derivatives—more complex than other product rules
Used in fluid vorticity equations—describes how rotation evolves in a flow

Compare: Divergence of cross product vs. Curl of cross product—divergence gives a scalar involving curls of the original fields, while curl gives a vector involving divergences and directional derivatives. The outputs have completely different structures.

Chain Rule and Composition

When vector fields depend on parameters or other functions, you need chain rules to differentiate properly.

Chain Rule for Vector-Valued Functions

$\frac{d\mathbf{F}}{dt} = \frac{\partial \mathbf{F}}{\partial x}\frac{dx}{dt} + \frac{\partial \mathbf{F}}{\partial y}\frac{dy}{dt} + \frac{\partial \mathbf{F}}{\partial z}\frac{dz}{dt}$ —sum over all variables
Compactly written as $\frac{d\mathbf{F}}{dt} = (\mathbf{v} \cdot \nabla)\mathbf{F}$ —where $\mathbf{v}$ is the velocity vector
Essential for material derivatives in fluid mechanics—tracks how quantities change following a moving particle

Quick Reference Table

Concept	Best Examples
Differential operators (basic)	Gradient $\nabla f$ , Divergence $\nabla \cdot \mathbf{F}$ , Curl $\nabla \times \mathbf{F}$
Second-order operators	Laplacian $\nabla^2 f$
Integral theorems	Green's, Stokes', Divergence Theorem
Product rules	$\nabla \cdot (\phi \mathbf{F})$ , $\nabla \times (\phi \mathbf{F})$
Triple products	BAC-CAB rule, Scalar triple product (volume)
Advanced identities	$\nabla(\mathbf{A} \cdot \mathbf{B})$ , $\nabla \cdot (\mathbf{A} \times \mathbf{B})$ , $\nabla \times (\mathbf{A} \times \mathbf{B})$
Zero results	$\nabla \times (\nabla f) = \mathbf{0}$ , $\nabla \cdot (\nabla \times \mathbf{F}) = 0$
Physical interpretation	Divergence = sources/sinks, Curl = rotation, Gradient = steepest ascent

Self-Check Questions

Which two identities would you use to show that $\nabla \cdot (\nabla \times \mathbf{F}) = 0$ for any smooth vector field $\mathbf{F}$ ?
Compare and contrast Stokes' theorem and the Divergence theorem: what type of integral does each convert, and what is the "boundary" in each case?
If a vector field $\mathbf{F}$ has zero divergence everywhere, what physical property does this indicate? Name one integral theorem that becomes especially useful for such fields.
You're given $\nabla \cdot (\phi \mathbf{F})$ and need to expand it. Write the product rule, then explain what happens if $\phi$ is constant.
An FRQ asks you to compute a line integral around a complicated closed curve in 3D. Which theorem could convert this to a surface integral, and what quantity would you integrate over the surface?