Multivariable Calculus

Year 1 · 5 sections · 56 concepts · 159 practice questions

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What's covered

free1 · Functions of several variables7 concepts

Overview 1 · The plan of this section
Definition 1.1 · Graphs and level sets
Definition 1.2 · Limits and continuity
Proposition 1.3 · The two-path test for non-existence
Definition 1.4 · Partial derivatives
Example 1.5 · Partial derivatives can exist where f is not continuous
Check 1.6 · Check: limits in two variables

2 · Differentiability13 concepts

Overview 2 · The plan of this section
Definition 2.1 · Differentiability and the tangent plane
Theorem 2.2 · Continuous partial derivatives imply differentiability
Proposition 2.3 · Differentiable implies continuous
Definition 2.4 · The gradient
Theorem 2.5 · Directional derivatives and steepest ascent
Remark 2.6 · Rates along a path
Proposition 2.7 · The gradient is normal to level sets
Theorem 2.8 · The multivariable chain rule
Definition 2.9 · The derivative matrix (Jacobian)
Technique 2.10 · Implicit differentiation and the implicit function theorem
Technique 2.11 · Linear approximation and error estimates
Check 2.12 · Check: multivariable derivatives

3 · Taylor and stationary points10 concepts

Overview 3 · The plan of this section
Definition 3.1 · Higher partial derivatives
Theorem 3.2 · Symmetry of mixed partials (Clairaut–Schwarz)
Theorem 3.3 · Second-order Taylor expansion and the Hessian
Definition 3.4 · Stationary points
Theorem 3.5 · The second derivative test
Technique 3.6 · Global extrema on a closed bounded region
Theorem 3.7 · Lagrange multipliers
Example 3.8 · Applications of Lagrange multipliers
Check 3.9 · Check: optimisation in several variables

4 · Multiple integrals12 concepts

Overview 4 · The plan of this section
Definition 4.1 · Double integrals
Theorem 4.2 · Fubini's theorem
Technique 4.3 · Integrating over general regions
Technique 4.4 · Changing the order of integration
Theorem 4.5 · Double integrals in polar coordinates
Example 4.6 · The Gaussian integral
Definition 4.7 · Triple integrals
Technique 4.8 · Cylindrical and spherical coordinates
Theorem 4.9 · Change of variables and the Jacobian
Example 4.10 · Mass and centre of mass
Check 4.11 · Check: setting up multiple integrals

5 · Vector calculus14 concepts

Overview 5 · The plan of this section
Definition 5.1 · Parametrised curves
Definition 5.2 · Vector fields
Definition 5.3 · Line integrals
Theorem 5.4 · The fundamental theorem for line integrals
Theorem 5.5 · Testing whether a field is conservative
Technique 5.6 · Finding a potential
Definition 5.7 · Divergence and curl
Proposition 5.8 · curl grad = 0 and div curl = 0
Theorem 5.9 · Green's theorem
Corollary 5.10 · Areas from line integrals
Definition 5.11 · Surface integrals and flux
Theorem 5.12 · The divergence theorem and Stokes' theorem (statements)
Check 5.13 · Check: line integrals and Green's theorem

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Proposition 1.3 · The two-path test for non-existence

Two paths, two answers ⇒ no limit. Lines aren't enough: $\frac{x^2y}{x^4 + y^2}$ needs the parabola $y = x^2$ . Typical paths: axes, $y = mx$ , $y = x^2$ .

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