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Formula sheet

Multivariable calculus — Formula Sheet

6 formulas from Multivariable Calculus, each with what it means, when to use it and the trap it sets.

Partial derivative

\frac{\partial f}{\partial x} : differentiate in x, treat y as a constant

What it says: The slope in one direction only, with every other variable frozen.
When to reach for it: Any function of more than one variable.
Watch out: A frozen variable is a constant, so its derivative is 0 — but it still multiplies anything it is attached to.

f = x^{2} y \Rightarrow f_{x} = 2 x y, f_{y} = x^{2}

In fₓ the y just came along for the ride.

Gradient

\nabla f = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z})

What it says: The direction of steepest increase, with length equal to that steepest rate.
When to reach for it: Optimisation, normal vectors to level surfaces, and every directional derivative.
Watch out: The gradient is a vector, and it is perpendicular to the level curve, not tangent to it.

\nabla (x^{2} + y^{2}) = (2 x, 2 y)

It points radially outward — straight uphill on a bowl.

Directional derivative

D_{u} f = \nabla f \cdot u^

What it says: The rate of change along a chosen direction.
When to reach for it: A question that names a direction rather than an axis.
Watch out: The direction must be normalised first. Using an unnormalised vector inflates the answer by its length.

D_{u^} f = \nabla f \cdot \frac{( 1 , 1 )}{2}

The √2 is the whole reason to normalise.

Tangent plane

z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)

What it says: The tangent line idea, with one term for each input direction.
When to reach for it: Linear approximation of a surface near a point.
Watch out: Both partials are evaluated at the point before being used as coefficients.

z = 1 + 2 (x - 1) + 2 (y - 1)

Tangent to z = x² + y² at (1,1).

Second derivative test

D = f_{xx} f_{y y} - f_{x y}^{2}

What it says: D > 0 with fₓₓ > 0 is a minimum, D > 0 with fₓₓ < 0 a maximum, D < 0 a saddle.
When to reach for it: Classifying a critical point of a surface.
Watch out: The cross term is squared and subtracted. D = 0 is inconclusive and needs another approach.

f = x^{2} - y^{2} : D = - 4 < 0

A saddle — a maximum one way and a minimum the other.

Polar double integral

\iint f d A = \iint f (r cos θ, r sin θ) r d r d θ

What it says: Switching to polar coordinates costs an extra factor of r.
When to reach for it: Circular or annular regions, and integrands containing x² + y².
Watch out: The extra r is not optional — it is the Jacobian, and forgetting it is the classic error.

\iint_{x^{2} + y^{2} \leq 1} d A = \int_{0}^{2 π} \int_{0}^{1} r d r d θ = π

Without the r this returns 2π and the area of the unit disc is not 2π.

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