Why Multivariable Calculus?
Most real phenomena depend on more than one variable. Temperature depends on x, y, z position. Profit depends on price and quantity. A surface in 3D space is described by z=f(x,y). Multivariable calculus extends the derivative and integral to handle all of this.
Functions of Two Variables
f(x,y) takes two inputs and produces one output. Its graph is a surface in 3D space. The domain is a region in the xy-plane. Level curves are obtained by setting f(x,y)=c — they show cross-sections of the surface at constant heights.
Partial Derivatives
The partial derivative ∂f/∂x treats y as a constant and differentiates with respect to x only. ∂f/∂y treats x as constant. Example: f(x,y)=x²y+3y². ∂f/∂x = 2xy. ∂f/∂y = x²+6y. Each partial derivative is a derivative in one direction with the other variable frozen.
The Gradient Vector
∇f = (∂f/∂x, ∂f/∂y). The gradient points in the direction of steepest ascent on the surface. Its magnitude is the rate of steepest increase. The gradient is perpendicular to the level curves — a fundamental fact used in optimisation and physics.
Every real-world quantity depends on more than one variable. Temperature, pressure, profit, neural network loss — all are functions of multiple inputs. Multivariable calculus provides the tools to analyse and optimise them.
Making the Leap from One to Many Variables
Single-variable calculus studies functions f: ℝ → ℝ. The graph is a curve in 2D. The derivative is a single number (slope). The integral is a single number (area). Multivariable calculus studies functions f: ℝⁿ → ℝ (or ℝᵐ). The graph is a surface or hypersurface. The "derivative" becomes a vector (the gradient) or a matrix (the Jacobian). The integral becomes a double, triple, or n-tuple integral over regions in space.
The core ideas — limit, derivative as rate of change, integral as accumulation — all carry over. What changes is the geometry and the notation. And crucially, new phenomena appear that have no single-variable analogue: saddle points, directional derivatives, line integrals, and the interplay of multiple rates of change.
Functions of Two Variables — Visualisation
f(x, y) = x² + y² is a paraboloid — a bowl shape opening upward. f(x, y) = x² − y² is a saddle surface — curving up in one direction, down in another. f(x, y) = sin(x)·cos(y) is a wavy surface resembling an egg carton. Visualising these surfaces is the first step in multivariable calculus. Level curves — obtained by setting f(x,y) = c for various constants c — are the 2D analogue of topographic contour lines. They show the height of the surface at every (x,y) point.
Partial Derivatives — Computing Them
For f(x,y) = x³y² + 5x − 2y:
- ∂f/∂x: treat y as a constant. ∂f/∂x = 3x²y² + 5.
- ∂f/∂y: treat x as a constant. ∂f/∂y = 2x³y − 2.
All the single-variable rules (power, product, chain, quotient) apply, because when you differentiate with respect to x, everything involving only y is treated as a constant coefficient. The Chain Rule for partial derivatives is more involved — it accounts for all paths through which x affects f.
The Gradient and Directional Derivatives
The gradient ∇f = (∂f/∂x, ∂f/∂y) is the multivariable analogue of the single-variable derivative. It points in the direction of steepest ascent. The rate of change of f in the direction of unit vector û is the directional derivative: D_û f = ∇f · û (dot product). This is maximum when û points along ∇f (the gradient direction), giving rate |∇f|. It is zero when û is perpendicular to ∇f (moving along a level curve changes f by zero). It is most negative when û points opposite ∇f.
Critical Points in Two Variables
A critical point of f(x,y) occurs where ∂f/∂x = 0 AND ∂f/∂y = 0 simultaneously. At a critical point, the gradient is the zero vector — no direction is uphill or downhill. Critical points can be local minima, local maxima, or saddle points (a minimum in one direction and maximum in another). The second derivative test uses the discriminant D = fₓₓfᵧᵧ − (fₓᵧ)²: D > 0 and fₓₓ > 0 → local minimum; D > 0 and fₓₓ < 0 → local maximum; D < 0 → saddle point.
Constrained Optimisation — Lagrange Multipliers
To optimise f(x,y) subject to a constraint g(x,y) = c, the method of Lagrange multipliers finds critical points of the Lagrangian. At the optimum, ∇f = λ∇g for some scalar λ (the Lagrange multiplier). Geometrically: the level curves of f and the constraint curve g = c are tangent at the optimal point. Solving the system ∂f/∂x = λ(∂g/∂x), ∂f/∂y = λ(∂g/∂y), g(x,y) = c gives the optimal point(s). This technique is used everywhere from economics (utility maximisation under a budget constraint) to machine learning (regularised optimisation).
From Single to Multiple Variables — A Comparison Table
- Derivative: f'(x) [scalar] → gradient ∇f [vector]
- Critical points: f'(c) = 0 → ∇f = 0 (both partials zero)
- Second derivative test: f''(c) sign → discriminant D = fₓₓfᵧᵧ − fₓᵧ²
- Integration: ∫f(x)dx [over interval] → ∬f(x,y)dA [over region]
- Chain Rule: dy/dx = (dy/du)(du/dx) → dz/dt = (∂z/∂x)(dx/dt) + (∂z/∂y)(dy/dt)
- Stewart, J. (2015). Multivariable Calculus, 8th ed. Cengage.
- Marsden, J. & Tromba, A. (2011). Vector Calculus, 6th ed. Freeman.
- Apostol, T. (1969). Calculus, Vol. 2. Wiley.