Unit 4: Multivariate differentiation

MTH165 — Mathematics For Engineers 6 min read

I. Foundations and Geometric Framework

Multivariate differentiation extends single-variable calculus to functions whose values depend on two or more independent variables. For a scalar function (z=f(x,y)), the graph is generally a surface, and differentiation describes its directional rates of change, tangent plane, local extrema, and response to constraints.

Domain and range: A function (f:D\subseteq\mathbb{R}^n\to\mathbb{R}) assigns a scalar (f(\mathbf{x})) to each point (\mathbf{x}=(x_1,\ldots,x_n)) in its domain (D).
Neighbourhood: A point (\mathbf{x}) approaches (\mathbf{a}) through infinitely many paths, so multivariable limits must be independent of the chosen path.
Increment notation: For (z=f(x,y)), changes (x\to x+\Delta x) and (y\to y+\Delta y) produce
TEXT
```
  Δz = f(x + Δx, y + Δy) − f(x,y),
```
where (\Delta z) is the total change in the dependent variable.
Geometric interpretation: First derivatives determine slopes and tangent planes; second derivatives measure curvature and help classify stationary points.
Regularity assumption: Continuity of the required partial derivatives near a point is a common sufficient condition for differentiability there, though partial derivatives alone do not guarantee differentiability.
Engineering relevance: Multivariable models describe quantities such as temperature (T(x,y,z)), pressure (P(V,T)), displacement, energy, cost, and constrained design performance.

II. Limits and Continuity — Local Behaviour of Multivariable Functions

A. Limits and continuity

Limits describe the value approached by a multivariable function near a point, while continuity requires that this limiting value equal the function value.

Formal limit: The statement
TEXT
```
  lim            f(x,y) = L
  (x,y) → (a,b)
```
means that for every (\varepsilon>0), there exists (\delta>0) such that
TEXT
```
  0 < √[(x−a)² + (y−b)²] < δ  ⇒  |f(x,y)−L| < ε.
```
Here ((a,b)) is the approached point, (L) is the limit, and (\varepsilon,\delta) are positive tolerances.
Path independence: A finite limit exists only if (f(x,y)) approaches the same (L) along every path to ((a,b)).
- Different limits along two paths prove non-existence.
- Equal limits along several tested paths do not by themselves prove existence.
Continuity at a point: The function (f) is continuous at ((a,b)) when:
TEXT
```
  f(a,b) is defined,
  lim f(x,y) exists,
  lim f(x,y) = f(a,b).
```
Standard continuous functions: Polynomials are continuous everywhere; rational functions are continuous wherever their denominators are non-zero. Sums, products, quotients with non-zero denominators, and compositions of continuous functions remain continuous.
Worked example: Consider
TEXT
```
  f(x,y) = xy/(x²+y²),   (x,y) ≠ (0,0).
```
Along (y=x), (f(x,x)=1/2). Along (y=-x), (f(x,-x)=-1/2). Since the path limits differ, the limit at ((0,0)) does not exist, regardless of any assigned value (f(0,0)).
Polar-coordinate test: Near the origin, set (x=r\cos\theta), (y=r\sin\theta). If the expression tends uniformly to a value independent of (\theta) as (r\to0), this can establish the limit.

III. Differential Structure — Partial and Total Change

A. Partial derivatives and total derivative

Partial derivatives isolate change in one variable, whereas the total derivative combines the simultaneous effects of all independent variables.

First partial derivatives: For (z=f(x,y)),

TEXT

  fₓ = ∂f/∂x = lim [f(x+h,y)−f(x,y)]/h,
                     h→0

  fᵧ = ∂f/∂y = lim [f(x,y+k)−f(x,y)]/k.
                     k→0

In (f_x), (y) is held constant; in (f_y), (x) is held constant.

Higher partial derivatives: Second-order derivatives are (f{xx}), (f{yy}), (f{xy}), and (f{yx}). If the mixed partials are continuous near a point, Clairaut’s theorem gives
TEXT
```
  fₓᵧ = fᵧₓ.
```
Gradient: The first partial derivatives form
TEXT
```
  ∇f = (fₓ, fᵧ),
```
where (\nabla f) points in the direction of steepest increase and (\lVert\nabla f\rVert) is the maximum directional rate of change.
Total derivative: If (f) is differentiable, its differential is
TEXT
```
  dz = fₓ dx + fᵧ dy.
```
Here (dx) and (dy) are small independent changes, and (dz) is the corresponding linear approximation to (\Delta z).
Differentiability criterion: A function is differentiable at ((a,b)) if
TEXT
```
  Δz = fₓ(a,b)Δx + fᵧ(a,b)Δy + R,
```
where (R/\sqrt{(\Delta x)^2+(\Delta y)^2}\to0). Continuous first partial derivatives near ((a,b)) are sufficient for this condition.
Worked example: For (f(x,y)=x^2y+3y),
TEXT
```
  fₓ = 2xy,       fᵧ = x²+3,
  dz = 2xy dx + (x²+3)dy.
```
At ((1,2)), (dz=4\,dx+4\,dy), which approximates the output change for small input changes.

IV. Chain Rule — Differentiation Through Dependence

A. Chain rule

The chain rule differentiates a composite function by adding the effects transmitted through every intermediate variable.

One-parameter dependence: If (z=f(x,y)), (x=x(t)), and (y=y(t)), then
TEXT
```
  dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt).
```
Each term represents one path by which (t) changes (z).

Two-parameter dependence: If (x=x(u,v)) and (y=y(u,v)), then

TEXT

  ∂z/∂u = fₓxᵤ + fᵧyᵤ,
  ∂z/∂v = fₓxᵥ + fᵧyᵥ.

Jacobian form: For vector-valued compositions, derivatives multiply in dependency order:
TEXT
```
  J_(f∘g)(u) = J_f(g(u)) J_g(u),
```
where (J) denotes the matrix of first partial derivatives.
Worked example: Let (z=x^2+y^2), (x=t^2), and (y=3t). Since (z_x=2x) and (z_y=2y),
TEXT
```
  dz/dt = 2x(2t) + 2y(3)
        = 4t³ + 18t.
```
This agrees with directly differentiating (z=t^4+9t^2).
Engineering use: The rule propagates sensitivities through dependent quantities, coordinate transformations, thermodynamic state variables, and multistage computational models.

V. Homogeneous Functions — Scaling and Euler’s Identity

A. Euler's theorem for homogeneous functions

Euler’s theorem connects a function’s scaling degree with a weighted sum of its first partial derivatives.

Homogeneous function: A function (f(x,y)) is homogeneous of degree (n) if
TEXT
```
  f(λx, λy) = λⁿf(x,y)
```
for every admissible scale factor (\lambda). For example, (x^2+xy+y^2) has degree (2).
Theorem: If (f) is differentiable and homogeneous of degree (n), then
TEXT
```
  x(∂f/∂x) + y(∂f/∂y) = nf(x,y).
```
Derivation: Define (F(\lambda)=f(\lambda x,\lambda y)=\lambda^nf(x,y)). Differentiating with respect to (\lambda) gives
TEXT
```
  x fₓ(λx,λy) + y fᵧ(λx,λy)
  = nλⁿ⁻¹f(x,y).
```
Setting (\lambda=1) yields Euler’s identity.
Extension: For (f(x_1,\ldots,x_m)) homogeneous of degree (n),
TEXT
```
  Σ xᵢ(∂f/∂xᵢ) = nf.
```
Worked example: For (f=x^2+3xy+2y^2), (f_x=2x+3y) and (f_y=3x+4y). Therefore,
TEXT
```
  xfₓ + yfᵧ = 2x²+6xy+4y² = 2f,
```
confirming degree (n=2).

VI. Unconstrained Optimization — Stationary Points and Curvature

A. Maxima and minima for a function of two variables

Local extrema occur where first-order change vanishes, while second derivatives determine the local curvature type.

Stationary-point condition: For an interior differentiable extremum of (f(x,y)),
TEXT
```
  fₓ(a,b) = 0,    fᵧ(a,b) = 0.
```
These conditions are necessary but not sufficient.
Second-derivative test: At a stationary point, define
TEXT
```
  D = fₓₓfᵧᵧ − (fₓᵧ)².
```
Then:
1. (D>0) and (f_{xx}>0): strict local minimum.
2. (D>0) and (f_{xx}<0): strict local maximum.
3. (D<0): saddle point.
4. (D=0): test is inconclusive.
Worked example: For
TEXT
```
  f(x,y) = x² + y² − 4x − 6y,
```
(f_x=2x-4) and (fy=2y-6), giving ((2,3)). Since (f{xx}=2), (f{yy}=2), (f{xy}=0), and (D=4>0), the point is a local minimum; completing squares shows it is also global.
Boundary limitation: On a closed bounded region, absolute extrema may occur at interior stationary points or on the boundary, so both must be checked.

VII. Constrained Optimization — Lagrange Multipliers

A. Lagrange method of multiplier

The Lagrange multiplier method finds extrema of an objective function while enforcing one or more equality constraints.

Principle: To optimize (f(x,y)) subject to (g(x,y)=c), define
TEXT
```
  ℒ(x,y,λ) = f(x,y) − λ[g(x,y)−c],
```
where (\lambda) is the Lagrange multiplier.
Necessary equations: At a regular constrained extremum,
TEXT
```
  fₓ = λgₓ,
  fᵧ = λgᵧ,
  g(x,y) = c.
```
Equivalently, (\nabla f=\lambda\nabla g): the objective and constraint gradients are parallel.
Geometric meaning: A constraint (g=c) is a level curve. At an extremum along this curve, its tangent direction produces no first-order change in (f), so both gradients are normal to the same tangent.
Worked example: Optimize (f(x,y)=xy) subject to (x+y=10). With (g=x+y),
TEXT
```
  y = λ,    x = λ,    x+y = 10.
```
Thus (x=y=5), giving (f=25). Under the usual positive-variable interpretation, this is the maximum product.
Multiple constraints: For constraints (g_i(\mathbf{x})=c_i),
TEXT
```
  ∇f = Σ λᵢ∇gᵢ,
```
with one multiplier (\lambda_i) for each independent constraint.
Limitations: Candidate points must still be compared, and singular points where (\nabla g=\mathbf{0}), endpoints, inequalities, or non-smooth boundaries require separate treatment.