Single-variable calculus asks how one output changes as one input moves. Multivariable calculus asks what “change” means when there are many possible directions.
This guide extends the rigorous R25 route from one dimension into Rⁿ. The shift is not merely “more variables”. New objects appear because a scalar derivative is no longer enough: gradients describe steepest first-order change, Jacobian matrices encode derivatives of vector-valued maps, Hessians organise second-order information, and determinants measure local volume distortion under coordinate changes.
Level: university-entry / undergraduate multivariable calculus and analysis enrichment. Prerequisites: limits and continuity, differentiability and local approximation, basic vectors and matrices, and single-variable integration.
Reading route: geometry of Rⁿ → limits → partial derivatives → total differentiability → directional derivatives → gradient → tangent planes → chain rule → Jacobian → inverse/implicit viewpoints → Hessian → Taylor approximation → unconstrained optimisation → Lagrange multipliers → double and triple integrals → iterated integration → change of variables → Jacobian determinant → vector fields → theorem boundaries → practice → solutions.
1. Points in Rⁿ carry several independent directions
A point x∈Rⁿ has coordinates x=(x₁,…,xₙ). A small change is a vector h=(h₁,…,hₙ), not a single number.
For a scalar field f:Rⁿ→R, the question “how fast is f changing?” is incomplete until a direction is specified. Moving east, north or diagonally through the same point can produce different first-order changes.
2. Limits in several variables require control from every direction
We say lim_{x→a}f(x)=L if for every ε>0 there exists δ>0 such that
0<||x−a||<δ implies |f(x)−L|<ε.
The norm controls all coordinate directions simultaneously. Checking a few convenient paths can disprove a limit, but agreement along finitely many paths never proves a multivariable limit.
3. Worked non-limit: path dependence
Consider f(x,y)=xy/(x²+y²) for (x,y)≠(0,0).
Along y=x, f(x,x)=1/2. Along y=−x, f(x,−x)=−1/2.
Two paths approaching the same point give different output limits, so the two-variable limit at the origin does not exist.
The method is one-way: path disagreement disproves a limit; path agreement alone does not establish it.
4. Partial derivatives freeze all but one coordinate
The partial derivative with respect to x_i is
∂f/∂x_i(a)=lim_{h→0}[f(a+he_i)−f(a)]/h
when the limit exists, where e_i is the i-th coordinate vector.
For f(x,y)=x²y+sin y, f_x=2xy and f_y=x²+cos y. These measure change along coordinate axes only.
5. Existing partial derivatives do not automatically imply differentiability
In one variable, existence of f'(a) is differentiability. In several variables, having all coordinate partial derivatives at a point is weaker.
Differentiability requires one linear map to approximate the function for every small direction simultaneously. This distinction is one of the first major theorem boundaries in multivariable calculus.
6. Total differentiability is a linear approximation theorem
A scalar function f:Rⁿ→R is differentiable at a if there exists a linear map L:Rⁿ→R such that
f(a+h)=f(a)+L(h)+r(h), with r(h)/||h||→0.
For scalar-valued functions, L(h)=∇f(a)·h when the derivative exists. This is the direct multivariable extension of the local-linearisation viewpoint from R25.02.
7. Continuity of first partial derivatives is a useful sufficient condition
If the first partial derivatives exist in a neighbourhood of a and are continuous at a, then f is differentiable at a.
This theorem is sufficient, not logically necessary. A function can be differentiable even if some first partial derivative is not continuous nearby.
The practical lesson is to distinguish “a convenient theorem that guarantees differentiability” from “the definition of differentiability itself”.
8. Directional derivatives test specified directions
For a unit vector u, the directional derivative is D_u f(a)=lim_{t→0}[f(a+tu)−f(a)]/t.
If f is differentiable, then D_u f(a)=∇f(a)·u. Existence of directional derivatives in every direction still does not, by itself, force total differentiability without additional control.
9. The gradient packages first-order scalar change
For f:Rⁿ→R, ∇f=(∂f/∂x₁,…,∂f/∂xₙ).
If f is differentiable, ∇f(a) points in the direction of greatest first-order increase, and its norm gives the maximum directional derivative over unit directions.
By Cauchy–Schwarz, D_u f=∇f·u≤||∇f||, with equality when u points along the gradient.
10. Level sets meet the gradient orthogonally
If f(x,y,z)=c defines a smooth level surface and ∇f(a)≠0, then tangent directions v satisfy ∇f(a)·v=0.
Thus the gradient is normal to the level surface. This supplies the analytic bridge to the geometric tangent-space language developed later in BTT’s Smooth Manifolds and Maps guide.
11. Tangent planes are first-order models
For z=f(x,y), the tangent plane at (a,b,f(a,b)) is
z≈f(a,b)+f_x(a,b)(x−a)+f_y(a,b)(y−b).
The approximation is not merely geometric. It is the differentiability statement written in coordinates.
12. The Jacobian is the derivative of a vector-valued map
For F:Rⁿ→Rᵐ with components F₁,…,F_m, the derivative is represented in standard coordinates by the Jacobian matrix J_F(a)=[∂F_i/∂x_j].
The local approximation becomes F(a+h)≈F(a)+J_F(a)h. The Jacobian is therefore not just a table of partial derivatives; it is the linear map controlling first-order behaviour.
13. The multivariable chain rule is matrix multiplication
If F:Rⁿ→Rᵐ and G:Rᵐ→Rᵖ are differentiable, then J_{G∘F}(a)=J_G(F(a))J_F(a).
Order matters because linear maps compose in order. The rightmost Jacobian acts first on the input perturbation. This matrix form is the exact higher-dimensional analogue of multiplying scalar derivatives in the one-variable chain rule.
14. Worked chain-rule example
Let u=x²+y² and F(x,y)=e^u. Then ∇u=(2x,2y) and d(e^u)/du=e^u.
Therefore ∇F=e^{x²+y²}(2x,2y). The calculation can be understood as composition R²→R→R and the chain rule between the two derivative maps.
15. The inverse function theorem uses a nonsingular Jacobian
For F:Rⁿ→Rⁿ continuously differentiable near a, if det J_F(a)≠0, then F has a differentiable local inverse near a.
The derivative of the local inverse is J_{F⁻¹}(F(a))=[J_F(a)]⁻¹.
A nonzero determinant is a local condition. It does not prove that F is globally one-to-one across its entire domain.
16. The implicit function theorem solves constraints locally
Suppose F(x,y)=0 and F_y(a,b)≠0. Under suitable smoothness conditions, the equation can be solved locally as y=g(x) near (a,b).
Differentiating F(x,g(x))=0 gives g'(x)=−F_x/F_y. The theorem explains why some constraint curves behave locally like ordinary function graphs even when no convenient global formula exists.
17. The Hessian organises second-order scalar information
For f:Rⁿ→R, the Hessian is H_f=[∂²f/∂x_i∂x_j].
When second partial derivatives are continuous near the point, mixed partials agree and the Hessian is symmetric. Its quadratic form hᵀH_f(a)h measures the dominant second-order correction to the tangent-plane model.
18. Second-order Taylor approximation uses the Hessian
Under suitable smoothness conditions,
f(a+h)=f(a)+∇f(a)·h+(1/2)hᵀH_f(a)h+remainder.
The first-order term gives slope; the quadratic term gives local curvature. Higher-dimensional approximation therefore combines linear algebra with calculus rather than merely adding more derivative rules.
19. Critical points solve ∇f=0
If f is differentiable and has an unconstrained local maximum or minimum at an interior point a, then ∇f(a)=0.
This is necessary, not sufficient. A critical point can be a minimum, maximum, saddle point or a degenerate point requiring higher-order analysis.
20. Hessian definiteness classifies nondegenerate critical points
- If H is positive definite, the point is a strict local minimum.
- If H is negative definite, it is a strict local maximum.
- If H is indefinite, it is a saddle point.
- If H is semidefinite or singular, the second-derivative test may be inconclusive.
For two variables, D=f_xx f_yy−f_xy² is a coordinate shortcut for this quadratic-form classification.
21. Worked saddle: f(x,y)=x²−y²
∇f=(2x,−2y), so the origin is critical. The Hessian is diag(2,−2), which has one positive and one negative eigenvalue.
Along y=0, f=x²≥0. Along x=0, f=−y²≤0. The origin is therefore a saddle point. Different directions reveal opposite second-order behaviour.
22. Lagrange multipliers handle equality constraints
To optimise f subject to g(x)=c, a regular constrained extremum often satisfies ∇f=λ∇g.
Geometrically, the objective’s level surface is tangent to the constraint surface at the optimum, so their normal vectors are parallel.
The condition generates candidates; it does not remove the need to inspect feasibility, endpoints, singular constraint points or global comparison.
23. Worked constrained optimisation
Maximise f(x,y)=xy subject to x²+y²=1.
∇f=(y,x), ∇g=(2x,2y). The equations y=2λx and x=2λy imply x²=y² when x,y are nonzero.
With x²+y²=1, |x|=|y|=1/√2. The maximum xy=1/2 occurs when x and y have the same sign; the minimum −1/2 occurs when their signs differ.
24. Double integrals accumulate over regions
For f defined on a region R⊂R², the double integral is the limit of sums Σ f(ξ_i)ΔA_i as the region is partitioned into smaller pieces under the appropriate integrability conditions.
If f≥0, the integral can represent volume under z=f(x,y). In general it is signed accumulation over area.
25. Fubini converts suitable multiple integrals into iterated integrals
For continuous f on a rectangle [a,b]×[c,d],
∫∫_R f(x,y)dA=∫_a^b[∫_c^d f(x,y)dy]dx
and the order may be reversed. More general Fubini and Tonelli theorems belong naturally to measure theory; their hypotheses matter, especially for sign-changing nonabsolute integrability.
26. Worked double integral on a rectangle
Compute ∫₀¹∫₀²(x+y)dy dx.
Inner integral: ∫₀²(x+y)dy=2x+2. Outer integral: ∫₀¹(2x+2)dx=3.
The result can be checked by reversing order and obtaining the same value.
27. Triple integrals extend accumulation into volume
For a density ρ(x,y,z), mass=∭_V ρ dV.
Centres of mass, moments and probability densities use the same accumulation architecture with different interpretations. Changing coordinates can make the geometry simpler, but the volume element must then be transformed correctly.
28. Change of variables introduces the Jacobian determinant
If T maps u-coordinates to x-coordinates smoothly and appropriately one-to-one, then locally |det J_T(u)| measures first-order area or volume scaling.
The change-of-variables formula is ∫_{T(U)} f(x)dx = ∫_U f(T(u))|det J_T(u)|du under the standard hypotheses.
29. Polar coordinates explain the factor r
Set x=r cosθ and y=r sinθ. The Jacobian determinant is r, so dA=r dr dθ.
The factor r is not an arbitrary rule. A small rectangle dr×dθ in parameter space becomes approximately a sector-shaped patch whose area scales like r dr dθ.
30. Worked polar integral: area of a disk
For the disk x²+y²≤R², area=∫₀^{2π}∫₀^R r dr dθ=πR².
The familiar area formula emerges from a coordinate transformation plus its local scaling determinant.
31. Vector fields prepare the bridge to vector calculus
A vector field assigns a vector F(x) to each point. Examples include velocity, force and flux fields. Gradient fields are special vector fields arising from scalar potentials.
Divergence measures local source-like expansion, while curl measures local rotational tendency in three-dimensional vector calculus. Line, surface and volume integral theorems—Green, Stokes and divergence—extend the local-to-global pattern. BTT’s Differential Forms and Integration guide develops the coordinate-independent version.
32. Common theorem-boundary failures
- Checking only coordinate-axis paths does not prove a multivariable limit.
- Existence of partial derivatives does not by itself prove differentiability.
- ∇f=0 identifies critical candidates, not guaranteed extrema.
- A nonsingular Jacobian gives local inverse structure, not global injectivity.
- Lagrange multiplier equations can miss singular constraint points and do not replace global comparison.
- Changing coordinates without the Jacobian determinant changes the quantity being integrated.
- Changing the order of improper or nonabsolutely integrable multiple integrals can fail without stronger hypotheses.
33. A dependable multivariable workflow
Identify the map type first: scalar field Rⁿ→R, vector-valued map Rⁿ→Rᵐ, or constrained relation.
For local change, distinguish coordinate partials, directional derivatives and total differentiability. Use the gradient only for scalar outputs; use the full Jacobian for vector outputs.
For optimisation, find interior critical points, inspect Hessian information when appropriate, and include boundaries or constraints.
For multiple integration, define the region, choose an order or coordinate system, and include the correct transformed volume factor.
At every stage, state whether the theorem is local or global and whether continuity, differentiability, compactness or integrability hypotheses have actually been checked.
34. Independent practice: twenty questions
- State the ε–δ definition of a multivariable limit using a norm.
- Use two paths to show xy/(x²+y²) has no limit at the origin.
- Find f_x and f_y for f=x²y+sin y.
- Why do existing partial derivatives not automatically imply differentiability?
- State the linear-map definition of differentiability.
- For differentiable f, express D_u f in terms of ∇f.
- What direction gives the greatest first-order increase of f?
- Why is ∇f normal to a regular level surface?
- Write the tangent-plane approximation for z=f(x,y) at (a,b).
- Define the Jacobian matrix of F:Rⁿ→Rᵐ.
- State the Jacobian form of the chain rule.
- What local condition appears in the inverse function theorem?
- What does the Hessian record?
- Classify the origin for f=x²−y².
- State the Lagrange multiplier condition for g=c.
- Optimise xy subject to x²+y²=1.
- What does Fubini’s theorem permit for continuous functions on rectangles?
- Why does polar integration contain a factor r?
- Compute the area of a disk of radius R by polar integration.
- Name two ways a multivariable theorem can fail when a hypothesis is dropped.
35. Worked solutions and checks
1. For every ε>0 there exists δ>0 such that 0<||x−a||<δ implies |f(x)−L|<ε.
2. Along y=x the value is 1/2; along y=−x it is −1/2, so no common limit exists.
3. f_x=2xy and f_y=x²+cos y.
4. They test only coordinate directions; differentiability requires one linear approximation controlling all small directions.
5. f(a+h)=f(a)+L(h)+r(h), where L is linear and ||r(h)||/||h||→0.
6. D_u f(a)=∇f(a)·u.
7. The unit direction parallel to ∇f, when the gradient is nonzero.
8. Tangent directions preserve the level value to first order, so ∇f·v=0.
9. z≈f(a,b)+f_x(a,b)(x−a)+f_y(a,b)(y−b).
10. J_F=[∂F_i/∂x_j].
11. J_{G∘F}(a)=J_G(F(a))J_F(a).
12. For F:Rⁿ→Rⁿ, det J_F(a)≠0 under the theorem’s smoothness hypotheses.
13. Second partial derivatives and the quadratic second-order response.
14. Saddle point; Hessian eigenvalues have opposite signs.
15. ∇f=λ∇g at regular constrained extrema, together with g=c.
16. Maximum 1/2 at points with |x|=|y|=1/√2 and matching signs; minimum −1/2 with opposite signs.
17. It allows a suitable double integral to be computed as iterated integrals and, under its hypotheses, permits reversing order.
18. The polar coordinate map has Jacobian determinant r, the local area scaling factor.
19. ∫₀^{2π}∫₀^R r dr dθ=πR².
20. Examples include path agreement not proving a limit, partials without differentiability, local inversion without global injectivity, or changing integration order without integrability control.
36. Where this guide hands off
R25.06 returns to the foundations of the real numbers themselves: least-upper-bound completeness, Cauchy convergence, nested intervals and the web of equivalent principles that make real analysis work. R25.07 then replaces rectangle-based Riemann accumulation with measure and Lebesgue integration.
Sources and further study
For a full undergraduate course route, see MIT OpenCourseWare 18.02 Multivariable Calculus, covering vectors, matrices, partial derivatives, multiple integrals and vector calculus. For geometric continuation, use BTT’s differential-geometry learning-guide batches.