Problem-Solving: Multivariable Calculus

Multivariable calculus problems require matching the right technique to the structure of the problem.

Decision guide

Task	Tool
Rate of change in a direction	Directional derivative $D_{\hat{u}}f$
Find extrema of $f$	$\nabla f = 0$ + second derivative test
Optimise with constraint	Lagrange multipliers
Integrate over a region	Double/triple integral
Circular/spherical symmetry	Change to polar/spherical coordinates
Line integral of conservative field	$\phi(B) - \phi(A)$
Closed-curve line integral	Green's theorem

Problem-solving checklist

Sketch the region or curve.
Identify the symmetry — choose coordinates accordingly.
Set up the integral with correct limits before computing.
Verify dimensions and limiting cases.

import numpy as np
from scipy import integrate
def f(y, x):
    return x**2 + y**2
result, _ = integrate.dblquad(f, 0, 1, 0, 1)
print(f"Integral of x^{2}+y^{2} over [0,1]x[0,1] = {result:.4f}")

Common mistakes

Wrong order of limits in iterated integrals.
Forgetting the Jacobian factor ( $r$ , $\rho^{2}\sin\phi$ ) when changing coordinates.
Applying Green's theorem to a non-closed curve.

Tip: Before computing, ask: is there a symmetry argument that makes the integral zero or reduces it? Exploit symmetry first, compute second.

Common pitfall: In multivariable problems, the decisive move happens before any integration: choosing coordinates that fit the region’s symmetry. Fighting a sphere with Cartesian limits is a self-inflicted wound.

Practise this lesson

Multivariable Calculus