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Mathematics · Advanced · 35 min · Chapter 9 of 28

Multivariable calculus

Functions of several variables, partial derivatives, the gradient, and integrals over regions of the plane and space.

Read first: Integral calculus, Linear algebra

The idea

When a quantity depends on several inputs, the derivative splits. The partial derivative ∂f/∂x is the ordinary derivative in x while y, z, and the rest are treated as constants. Geometrically, for z = f(x, y), it is the slope of the surface in the x-direction. Mixed partials ∂²f/∂y∂x and ∂²f/∂x∂y agree whenever they are continuous, which is Clairaut’s theorem and the reason you can usually differentiate in either order.

The gradient ∇f is the vector of all first partials. It is perpendicular to the level set of f, and the directional derivative in a unit direction u is the dot product ∇f · u. That product is largest when u points along ∇f, which is the precise meaning of steepest ascent. Setting the gradient to zero locates the critical points of a function of several variables; the Hessian, the matrix of second partials, plays the role of the second derivative test.

Integration adds a dimension of bookkeeping. ∬_D f(x, y) dA is the limit of sums f(xᵢ, yᵢ) ΔA over a region D. Iterated integrals compute it by sweeping one variable, then the other, and the limits on the inner integral may depend on the outer variable. In polar coordinates the area element is r dr dθ, not dr dθ — the factor r is the Jacobian, and forgetting it is the classic error. Divergence ∇ · F and curl ∇ × F package the derivatives of a vector field; the divergence theorem and Stokes’ theorem are the fundamental theorem in higher dimensions.

Hold onto this

  1. A partial derivative holds every variable but one fixed, then differentiates.
  2. The gradient points in the direction of steepest increase, and its magnitude is that slope.
  3. A double integral adds values over an area; a triple integral, over a volume.
  4. Divergence measures source, curl measures local rotation.

Formulas

Partial derivative

Differentiate with respect to x, holding y fixed.

∂
Partial derivative, not a fraction you cancel casually

Gradient

The vector of first partials. It points uphill most steeply.

∇
Del, or nabla

Directional derivative

Rate of change in the direction of a unit vector u.

u
A unit vector

Double integral

Signed volume under the surface z = f over the region D, or accumulated density if f is a density.

dA
Area element: dx dy, or r dr dθ in polar

Divergence and curl

Divergence is a scalar measuring net outflow. Curl is a vector measuring local rotation.

F
A vector field

Worked examples

1. Partials of a product surface

Let f(x, y) = x²y + sin y. Find both first partials and the gradient at (1, 0).

  1. ∂f/∂x = 2xy. At (1, 0) this is 0.
  2. ∂f/∂y = x² + cos y. At (1, 0) this is 1 + 1 = 2.
  3. Gradient at (1, 0) is the vector (0, 2).

∇f(1, 0) = (0, 2). The surface climbs in the y-direction and is instantaneously flat in x.

2. Area in polar coordinates

Find the area of the disk x² + y² ≤ 4 using a double integral.

  1. In polar, the disk is 0 ≤ r ≤ 2 and 0 ≤ θ ≤ 2π, and dA = r dr dθ.
  2. ∫ from 0 to 2π of ∫ from 0 to 2 of r dr dθ.
  3. Inner integral: r²/2 from 0 to 2 = 2.
  4. Outer: 2 × 2π = 4π.

4π, matching πR² with R = 2.

Try these

For f(x, y) = eˣʸ, compute ∂f/∂x.

In which direction from (1, 1) does f(x, y) = x² + 2y² increase fastest?

Common mistakes

  • In polar and spherical coordinates the volume element carries a Jacobian. dA is not always dx dy.
  • A partial derivative symbol is not an instruction to cancel ∂ against something else.
  • Critical points of functions of several variables include saddles. A zero gradient is not automatically a peak.