Unit 4 - Notes

MTH165

Unit 4: Multivariate functions

1. Functions of Two Variables

Definition

A real-valued function of two variables, $f: D \to \mathbb{R}$ , assigns a unique real number $z = f(x, y)$ to each ordered pair $(x, y)$ in a subset $D$ of $\mathbb{R}^2$ .

Independent Variables: $x$ and $y$ .
Dependent Variable: $z$ .

Domain and Range

Domain ( $D$ ): The set of all pairs for which the function is mathematically defined.
- Example: For $f(x, y) = \sqrt{9 - x^2 - y^2}$ , the domain is $x^2 + y^2 \leq 9$ (a disk of radius 3).
Range: The set of all possible output values $z$ .

Graphical Representation

Surface: The graph of a function $f(x, y)$ is a surface in 3-dimensional space (Euclidean space $\mathbb{R}^3$ ).
Level Curves (Contour Lines): The set of points in the -plane where (a constant).
- If you slice the surface $z = f(x, y)$ with horizontal planes $z=c$ , the intersection curves projected onto the $xy$ -plane are level curves.
- Closely spaced level curves indicate a steep slope; widely spaced curves indicate a gentle slope.

2. Limits and Continuity

Limit of a Function

The notation $\lim_{(x, y) \to (a, b)} f(x, y) = L$ means that values of $f(x, y)$ approach the number $L$ as the point $(x, y)$ approaches $(a, b)$ along any path within the domain.

The Two-Path Test (Non-existence of Limits)

Unlike single-variable calculus (where you check left and right limits), in 2D, you can approach a point from infinitely many directions.

Rule: If $f(x, y) \to L_1$ along Path A, and $f(x, y) \to L_2$ along Path B, and $L_1 \neq L_2$ , then the limit does not exist.
Common Paths to Check:
1. Along the x-axis ( $y=0$ ).
2. Along the y-axis ( $x=0$ ).
3. Along the line $y=x$ or $y=mx$ .
4. Along parabolas (e.g., $y=x^2$ ) if degrees imply it.

Continuity

A function $f(x, y)$ is continuous at a point $(a, b)$ if three conditions are met:

$f(a, b)$ is defined.
$\lim_{(x, y) \to (a, b)} f(x, y)$ exists.
$\lim_{(x, y) \to (a, b)} f(x, y) = f(a, b)$ .

Polynomial functions are continuous everywhere. Rational functions are continuous everywhere except where the denominator is zero.

3. Partial Derivatives

Definition

A partial derivative measures the rate of change of the function with respect to one variable while holding the other variable(s) constant.

Given $z = f(x, y)$ :

Partial with respect to $x$ :

$\frac{\partial f}{\partial x} = f_x(x, y) = \lim_{h \to 0} \frac{f(x+h, y) - f(x, y)}{h}$
(Treat $y$ as a constant constant).
Partial with respect to $y$ :

$\frac{\partial f}{\partial y} = f_y(x, y) = \lim_{k \to 0} \frac{f(x, y+k) - f(x, y)}{k}$
(Treat $x$ as a constant constant).

Geometric Interpretation

$f_x(a, b)$ is the slope of the tangent line to the curve $z = f(x, b)$ at $x=a$ (intersection of the surface with the plane $y=b$ ).
$f_y(a, b)$ is the slope of the tangent line to the curve $z = f(a, y)$ at $y=b$ (intersection of the surface with the plane $x=a$ ).

Higher-Order Partial Derivatives

Second order derivatives:
- $f_{xx} = \frac{\partial}{\partial x}(\frac{\partial f}{\partial x}) = \frac{\partial^2 f}{\partial x^2}$
- $f_{yy} = \frac{\partial}{\partial y}(\frac{\partial f}{\partial y}) = \frac{\partial^2 f}{\partial y^2}$
- Mixed Partials:
  - $f_{xy} = \frac{\partial}{\partial y}(\frac{\partial f}{\partial x}) = \frac{\partial^2 f}{\partial y \partial x}$
  - $f_{yx} = \frac{\partial}{\partial x}(\frac{\partial f}{\partial y}) = \frac{\partial^2 f}{\partial x \partial y}$

Clairaut's Theorem (Equality of Mixed Partials):
If $f_{xy}$ and $f_{yx}$ are both continuous on a disk containing $(a, b)$ , then $f_{xy}(a, b) = f_{yx}(a, b)$ .

4. Total Derivative and Differentiability

Total Differential

If $z = f(x, y)$ , the total differential $dz$ is defined as:

dz = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy

Usage: Used for approximation and error estimation. If $\Delta x$ and $\Delta y$ are small changes, the change in $z$ ( $\Delta z$ ) is approximated by:
$\Delta z \approx f_x \Delta x + f_y \Delta y$

Differentiability

Existence of partial derivatives does not guarantee differentiability.
A function $f(x, y)$ is differentiable at $(a, b)$ if the increment $\Delta z$ can be written as:

\Delta z = f_x(a, b)\Delta x + f_y(a, b)\Delta y + \epsilon_1 \Delta x + \epsilon_2 \Delta y

where

\epsilon_1, \epsilon_2 \to 0

(\Delta x, \Delta y) \to (0, 0)

Geometrically: If $f$ is differentiable at a point, the surface has a unique non-vertical tangent plane at that point.

Sufficient Condition for Differentiability: If partial derivatives $f_x$ and $f_y$ exist and are continuous near $(a, b)$ , then $f$ is differentiable at $(a, b)$ .

5. Chain Rule

The Chain Rule for multivariate functions depends on the dependency tree of variables.

Case 1: One Independent Variable

If $z = f(x, y)$ , where $x = g(t)$ and $y = h(t)$ :

\frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt}

Case 2: Two Independent Variables

If $z = f(x, y)$ , where $x = g(u, v)$ and $y = h(u, v)$ :

\frac{\partial z}{\partial u} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial u}

\frac{\partial z}{\partial v} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial v} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial v}

Implicit Differentiation

If $F(x, y) = 0$ defines $y$ implicitly as a function of $x$ :

\frac{dy}{dx} = -\frac{F_x}{F_y} \quad (\text{provided } F_y \neq 0)

6. Euler's Theorem for Homogeneous Functions

Homogeneous Function

A function $f(x, y)$ is homogeneous of degree $n$ if, for any constant $t > 0$ :

f(tx, ty) = t^n f(x, y)

Alternatively, a function can be written as $x^n \phi(\frac{y}{x})$ or $y^n \phi(\frac{x}{y})$ .

Euler's Theorem

If $u = f(x, y)$ is a homogeneous function of degree $n$ having continuous partial derivatives, then:

x \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial y} = n u

Corollary (Second Order)

For a homogeneous function of degree $n$ :

x^2 \frac{\partial^2 u}{\partial x^2} + 2xy \frac{\partial^2 u}{\partial x \partial y} + y^2 \frac{\partial^2 u}{\partial y^2} = n(n-1)u

7. Maxima and Minima (Unconstrained Optimization)

Critical Points

A point $(a, b)$ is a critical point (stationary point) if:

$f_x(a, b) = 0$ and $f_y(a, b) = 0$ , or
The partial derivatives do not exist.

Second Derivative Test

To classify a critical point $(a, b)$ where $f_x = f_y = 0$ :

Calculate second partials at $(a, b)$ : $A = f_{xx}$ , $B = f_{xy}$ , $C = f_{yy}$ .
Compute the discriminant (Hessian determinant):
$D = AC - B^2 = f_{xx}f_{yy} - (f_{xy})^2$

Classification Rules:

Local Minimum: If $D > 0$ and $A > 0$ (convex up).
Local Maximum: If $D > 0$ and $A < 0$ (concave down).
Saddle Point: If $D < 0$ (curve goes up in one direction and down in another, like a Pringles chip).
Inconclusive: If $D = 0$ (test fails; further investigation required).

8. Lagrange Method of Multipliers

Purpose

Used to find the local maxima and minima of a function $f(x, y)$ subject to a constraint $g(x, y) = k$ .

The Method

Formulate the Lagrangian Function:

$L(x, y, \lambda) = f(x, y) - \lambda [g(x, y) - k]$
Here, $\lambda$ is a new variable called the Lagrange Multiplier.
Determine the Gradient Conditions:
Solve the system of equations formed by setting the partial derivatives of $L$ to zero:
- $\frac{\partial L}{\partial x} = f_x - \lambda g_x = 0$
- $\frac{\partial L}{\partial y} = f_y - \lambda g_y = 0$
- $\frac{\partial L}{\partial \lambda} = -(g(x, y) - k) = 0 \implies g(x, y) = k$
Solve the System:
Solve the simultaneous equations for $x, y,$ and $\lambda$ .
Evaluate:
Evaluate $f(x, y)$ at all solution points to determine which are maximum or minimum values.

Geometric Interpretation

At the optimum point under constraint, the level curve of the function $f(x, y) = c$ is tangent to the constraint curve $g(x, y) = k$ . Therefore, their gradients are parallel:

\nabla f = \lambda \nabla g

Unit 4 - Notes

Table of Contents

Unit 4: Multivariate functions

1. Functions of Two Variables

Definition

Domain and Range

Graphical Representation

2. Limits and Continuity

Limit of a Function

The Two-Path Test (Non-existence of Limits)

Continuity

3. Partial Derivatives

Definition

Geometric Interpretation

Higher-Order Partial Derivatives

4. Total Derivative and Differentiability

Total Differential

Differentiability

5. Chain Rule

Case 1: One Independent Variable

Case 2: Two Independent Variables

Implicit Differentiation

6. Euler's Theorem for Homogeneous Functions

Homogeneous Function

Euler's Theorem

Corollary (Second Order)

7. Maxima and Minima (Unconstrained Optimization)

Critical Points

Second Derivative Test

8. Lagrange Method of Multipliers

Purpose

The Method

Geometric Interpretation