Multivariable Calculus Theorems

Why This Matters

These theorems aren't just abstract formulas—they're the connective tissue of vector calculus, linking line integrals, surface integrals, volume integrals, and derivatives into a coherent framework. You're being tested on your ability to recognize when to apply each theorem, how they relate to one another, and why certain conditions (like conservative fields or continuous partial derivatives) make computations dramatically simpler. The big picture? These results let you convert hard integrals into easier ones and reveal deep relationships between local behavior (derivatives, divergence, curl) and global behavior (integrals over boundaries).

Don't just memorize the theorem statements—know what type of problem each theorem solves and how they form a hierarchy. Green's Theorem is a special case of Stokes' Theorem, which connects to the Divergence Theorem through the broader framework of differential forms. When you see an integral, ask yourself: Can I convert this to a simpler domain? Is this field conservative? That conceptual reflex is what separates strong exam performance from mere formula recall.

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Fundamental Theorems Connecting Integrals and Derivatives

These theorems share a common structure: they relate an integral over a region to an integral over its boundary, reducing dimensional complexity and revealing when path independence applies.

Gradient Theorem (Fundamental Theorem of Line Integrals)

• Path independence for conservative fields—if $\mathbf{F} = \nabla f$, then $\int_C \mathbf{F} \cdot d\mathbf{r} = f(\mathbf{b}) - f(\mathbf{a})$
• Potential function evaluation replaces tedious parameterization; only endpoint values matter
• Conservative field test—if $\nabla \times \mathbf{F} = \mathbf{0}$ on a simply connected domain, the field has a potential function

Green's Theorem

• Circulation-curl form—relates $\oint_C \mathbf{F} \cdot d\mathbf{r}$ to $\iint_R \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA$ for planar regions
• Flux-divergence form converts $\oint_C \mathbf{F} \cdot \mathbf{n} \, ds$ to $\iint_R \nabla \cdot \mathbf{F} \, dA$
• Area computation trick—set up $\frac{1}{2} \oint_C (x \, dy - y \, dx)$ to find enclosed area via line integral

Stokes' Theorem

• Surface-boundary relationship—$\iint_S (\nabla \times \mathbf{F}) \cdot d\mathbf{S} = \oint_C \mathbf{F} \cdot d\mathbf{r}$ where $C = \partial S$
• Generalizes Green's Theorem to arbitrary oriented surfaces in $\mathbb{R}^3$; orientation must be consistent via right-hand rule
• Curl interpretation—measures local rotation; zero curl everywhere implies conservative field (on simply connected domains)

Divergence Theorem (Gauss's Theorem)

• Volume-surface relationship—$\iiint_V \nabla \cdot \mathbf{F} \, dV = \oiint_S \mathbf{F} \cdot d\mathbf{S}$ for closed surfaces
• Flux computation converts difficult surface integrals to volume integrals when divergence is simpler
• Physical interpretation—divergence measures source/sink strength; net outward flux equals total source inside

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Differentiation Rules for Multivariable Functions

These theorems establish how derivatives behave when functions depend on multiple variables, ensuring consistency and enabling chain-rule computations.

Chain Rule for Multivariable Functions

• Composite function differentiation—if $z = f(x, y)$ where $x = x(t)$ and $y = y(t)$, then $\frac{dz}{dt} = \frac{\partial f}{\partial x}\frac{dx}{dt} + \frac{\partial f}{\partial y}\frac{dy}{dt}$
• Tree diagram method tracks all dependency paths; sum contributions from each branch
• Jacobian matrices generalize this to vector-valued functions: $D(\mathbf{g} \circ \mathbf{f}) = D\mathbf{g} \cdot D\mathbf{f}$

Clairaut's Theorem (Symmetry of Mixed Partials)

• Order independence—$\frac{\partial^2 f}{\partial x \partial y} = \frac{\partial^2 f}{\partial y \partial x}$ when both mixed partials are continuous
• Continuity requirement is essential; counterexamples exist when this fails
• Practical use—simplifies Hessian matrix construction and verifies computation accuracy

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Approximation and Local Behavior

These results let you approximate complicated functions near a point and understand implicit relationships between variables.

Taylor's Theorem for Multivariable Functions

• Polynomial approximation—$f(\mathbf{x}_0 + \mathbf{h}) \approx f(\mathbf{x}_0) + \nabla f \cdot \mathbf{h} + \frac{1}{2}\mathbf{h}^T H \mathbf{h} + \cdots$ where $H$ is the Hessian
• Second-order terms involve the Hessian matrix of second partials; critical for classifying critical points
• Error bounds depend on higher derivatives over the region; useful for numerical analysis applications

Implicit Function Theorem

• Existence guarantee—if $F(x, y) = 0$ and $\frac{\partial F}{\partial y} \neq 0$ at a point, then $y = g(x)$ exists locally
• Derivative formula—$\frac{dy}{dx} = -\frac{\partial F/\partial x}{\partial F/\partial y}$ without solving explicitly for $y$
• Generalizes to systems: non-singular Jacobian (with respect to dependent variables) guarantees local solvability

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Optimization Under Constraints

This theorem provides the standard method for constrained optimization, appearing constantly in applications and exams.

Lagrange Multiplier Theorem

• Constraint incorporation—at constrained extrema, $\nabla f = \lambda \nabla g$ where $g(\mathbf{x}) = c$ is the constraint
• Geometric interpretation—optimal points occur where level curves of $f$ are tangent to the constraint surface
• Multiple constraints use multiple multipliers: $\nabla f = \lambda_1 \nabla g_1 + \lambda_2 \nabla g_2 + \cdots$

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Quick Reference Table

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Self-Check Questions

1. Which two theorems both relate circulation around a boundary to a "curl-type" integral, and what distinguishes their domains of application?

2. If you're given a vector field and asked whether a line integral is path-independent, which theorem justifies your answer, and what condition must you verify?

3. Compare the Divergence Theorem and Stokes' Theorem: what type of integral does each convert, and what differential operator appears in each?

4. An FRQ gives you $F(x, y, z) = 0$ and asks for $\frac{\partial z}{\partial x}$. Which theorem applies, and what must be nonzero for it to work?

5. You need to maximize $f(x, y)$ subject to $g(x, y) = k$. Write the system of equations you must solve, and explain geometrically why $\nabla f$ and $\nabla g$ must be parallel at the solution.