Unit 4: Multivariate differentiation

I. Orientation — Functions of Several Variables

Multivariate differentiation extends single-variable calculus to functions whose outputs 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 behavior under constraints.

• Domain and range: A function (f:D\subseteq\mathbb{R}^n\to\mathbb{R}) assigns a scalar (f(\mathbf{x})) to each vector (\mathbf{x}=(x_1,\ldots,x_n)) in its domain (D).
• Geometric interpretation: For (z=f(x,y)), fixing (y) produces a curve in the (xz)-direction, while fixing (x) produces one in the (yz)-direction.
• Increment convention: Changes in the inputs are denoted by (\Delta x,\Delta y), and the corresponding output change is
  
  TEXT

    Δz = f(x + Δx, y + Δy) − f(x, y).

  
  Here, ((x,y)) is the initial point and ((\Delta x,\Delta y)) is its displacement.
• Local viewpoint: Limits describe behavior near a point; derivatives give local linear approximations; optimization identifies points where nearby function values are larger or smaller.
• Regularity assumptions: Continuity does not necessarily imply differentiability, but continuous first partial derivatives near a point are sufficient for differentiability there.

II. Limits and Continuity — Local Behavior Near a Point

A. limits and continuity

A multivariable limit exists only when the function approaches the same value along every possible path to the specified point.

• Formal limit: The statement
  
  TEXT

    lim               f(x, y) = L
    (x, y) → (a, b)

  
  means that for every (\varepsilon>0), there is a (\delta>0) such that
  
  TEXT

    0 < √[(x − a)² + (y − b)²] < δ  ⇒  |f(x, y) − L| < ε.

  
  Here, ((a,b)) is the limiting point, (L) is the proposed limit, and (\varepsilon,\delta) are positive tolerances.
• Path requirement: Equal limits along the lines (y=mx) are necessary but not sufficient for existence, because curved paths such as (y=x^2) must also agree.
• Non-existence test: For
  
  TEXT

    f(x, y) = xy/(x² + y²),

  
  approach ((0,0)) along (y=x) to obtain (1/2), but along (y=-x) to obtain (-1/2). The unequal values prove that the limit does not exist.
• Continuity: The function (f) is continuous at ((a,b)) when (f(a,b)) is defined, the limit exists, and
  
  TEXT

    lim               f(x, y) = f(a, b).
    (x, y) → (a, b)

• Useful properties: Polynomial functions are continuous everywhere; rational functions are continuous wherever their denominators are nonzero; sums, products, quotients, and compositions preserve continuity wherever defined.

III. Differential Structure — Local Linear Change

A. partial derivatives and total derivative

Partial derivatives measure change in one independent variable at a time, whereas the total derivative combines all first-order changes into one linear approximation.

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

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

  
  Here, (h) and (k) are scalar increments; the other variable is held constant in each limit.
• Higher partial derivatives: Second derivatives include (f{xx}), (f{yy}), and the mixed derivatives (f{xy}) and (f{yx}). If the mixed derivatives are continuous near a point, then
  
  TEXT

    fₓᵧ = fᵧₓ.

• Gradient: The first partial derivatives form the vector
  
  TEXT

    ∇f = (fₓ, fᵧ).

  
  The gradient points in the direction of greatest increase, and its magnitude gives the maximum directional rate of change.
• Differentiability and total differential: If (f) is differentiable at ((x,y)), then
  
  TEXT

    Δz = fₓ Δx + fᵧ Δy + o(√[(Δx)² + (Δy)²]),
    dz = fₓ dx + fᵧ dy.

  
  The notation (o(r)) represents an error that becomes negligible relative to (r) as (r\to0); (dx,dy) are small input changes.
• Worked example: For (f(x,y)=x^2y+3y^2),
  
  TEXT

    fₓ = 2xy,     fᵧ = x² + 6y,
    df = 2xy dx + (x² + 6y)dy.

  
  At ((1,2)), with (dx=0.01) and (dy=-0.02), (df=4(0.01)+13(-0.02)=-0.22), approximating the actual change in (f).

IV. Chain Rule — Differentiating Composite Dependence

A. chain rule

The multivariable chain rule traces how an output changes through every intermediate variable on which it depends.

• One parameter: If (z=f(x,y)), with (x=x(t)) and (y=y(t)), then
  
  TEXT

    dz/dt = (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt).

  
  Here, (t) is the independent parameter and (x,y) are intermediate variables.
• Two parameters: If (x=x(u,v)) and (y=y(u,v)), then
  
  TEXT

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

  
  Subscripts denote partial differentiation with respect to the indicated variable.
• Jacobian form: For differentiable vector functions, composition becomes matrix multiplication:
  
  TEXT

    J(F ∘ G) = JF(G) · JG.

  
  Here, (F) is the outer function, (G) is the inner function, and (J) denotes the matrix of first partial derivatives.
• Worked example: Let (z=x^2+y^2), (x=t^2), and (y=3t). Then
  
  TEXT

    dz/dt = 2x(2t) + 2y(3)
          = 4t³ + 18t.

  
  This agrees with directly differentiating (z=t^4+9t^2).

V. Homogeneous Functions — Scaling and Euler’s Identity

A. Euler's theorem for homogeneous functions

Euler’s theorem converts a scaling property into a differential identity for a differentiable homogeneous function.

• Homogeneity: A function (f(x,y)) is homogeneous of degree (n) if
  
  TEXT

    f(λx, λy) = λⁿf(x, y)

  
  for every admissible scalar (\lambda). The number (n) is the degree of homogeneity.
• Euler’s theorem: If (f) is differentiable and homogeneous of degree (n), then
  
  TEXT

    x(∂f/∂x) + y(∂f/∂y) = nf(x, y).

• Derivation: Differentiate (f(\lambda x,\lambda y)=\lambda^nf(x,y)) with respect to (\lambda):
  
  TEXT

    x fₓ(λx, λy) + y fᵧ(λx, λy) = nλⁿ⁻¹f(x, y).

  
  Setting (\lambda=1) gives Euler’s identity.
• Worked example: For (f=x^3+3x^2y+y^3), every term has total degree (3), so (n=3). Direct differentiation gives
  
  TEXT

    xfₓ + yfᵧ
    = x(3x² + 6xy) + y(3x² + 3y²)
    = 3f.

• Limitation: Euler’s theorem requires a genuine scaling relation; a function such as (x^2+y) is not homogeneous because its terms have different degrees.

VI. Unconstrained Optimization — Stationary-Point Classification

A. maxima and minima for a function of two variables

Local extrema occur where nearby function values are no greater or no smaller, and interior differentiable extrema must be stationary points.

• Definitions:
  1. Local maximum: (f(a,b)\geq f(x,y)) for all ((x,y)) sufficiently close to ((a,b)).
  2. Local minimum: (f(a,b)\leq f(x,y)) in a sufficiently small neighborhood.
• Necessary condition: At an interior local extremum where (f) is differentiable,
  
  TEXT

    fₓ(a, b) = 0,     fᵧ(a, b) = 0.

  
  A solution ((a,b)) is a critical point, but it may instead be a saddle point.
• Second-derivative test: Compute
  
  TEXT

    D = fₓₓfᵧᵧ − (fₓᵧ)²

  
  at the critical point.
  • If (D>0) and (f_{xx}>0), there is a local minimum.
  • If (D>0) and (f_{xx}<0), there is a local maximum.
  • If (D<0), there is a saddle point.
  • If (D=0), the test is inconclusive.
• Worked example: For (f=x^2+y^2-4x+6y),
  
  TEXT

    fₓ = 2x − 4,     fᵧ = 2y + 6.

  
  Thus the only critical point is ((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 the global minimum.
• Boundary issue: On a closed bounded domain, absolute extrema may occur at interior critical points or on the boundary, so both must be checked.

VII. Constrained Optimization — The Multiplier Technique

A. Lagrange method of multiplier

The Lagrange multiplier method finds extrema of (f(x,y)) subject to a constraint (g(x,y)=c) by requiring their gradients to be parallel.

• Geometric principle: At a constrained extremum, the level curve of (f) is tangent to the constraint curve, giving
  
  TEXT

    ∇f = λ∇g.

  
  Here, (\lambda) is the Lagrange multiplier, (\nabla f=(f_x,f_y)), and (\nabla g=(g_x,g_y)).
• System to solve:
  
  TEXT

    fₓ = λgₓ,
    fᵧ = λgᵧ,
    g(x, y) = c.

  
  The equations determine candidate values of (x), (y), and (\lambda), provided (\nabla g\neq\mathbf{0}).
• Worked example: To extremize (f=xy) subject to (x^2+y^2=2), set
  
  TEXT

    y = 2λx,
    x = 2λy,
    x² + y² = 2.

  
  The nonzero solutions satisfy (x=\pm y). The candidates ((1,1)) and ((-1,-1)) give the maximum (f=1); ((1,-1)) and ((-1,1)) give the minimum (f=-1).
• Interpretation of (\lambda): Under suitable regularity conditions, (\lambda) measures the approximate change in the optimal value when the constraint level (c) changes.
• Limitations: Singular points where (\nabla g=\mathbf{0}), endpoints, corners, and multiple constraints require separate checks or an extended system with one multiplier per constraint.