Unit 5 - Notes

MTH174 7 min read

Unit 5: Multivariate Calculus

1. Limit, Continuity, and Differentiability of Functions of Two Variables

1.1 Functions of Two Variables

A function of two variables, usually denoted as $z = f(x, y)$ , is a rule that assigns a unique real number $z$ to every ordered pair $(x, y)$ in a specific set $D$ (the domain) in the $xy$ -plane.

1.2 Limits

The limit of a function $f(x, y)$ as $(x, y)$ approaches $(x_0, y_0)$ is $L$ , written as:
$\lim_{(x,y) \to (x_0, y_0)} f(x, y) = L$
This means that the value of $f(x, y)$ can be made as close to $L$ as desired by choosing $(x, y)$ sufficiently close to $(x_0, y_0)$ .

Crucial Point for Multivariable Limits:
For the limit to exist, $f(x, y)$ must approach $L$ along every possible path passing through $(x_0, y_0)$ . If you can find two different paths (e.g., along the x-axis, y-axis, or a line $y = mx$ ) that yield different limit values, the limit does not exist.

1.3 Continuity

A function $f(x, y)$ is continuous at a point $(x_0, y_0)$ if:

$f(x_0, y_0)$ is defined.
$\lim_{(x,y) \to (x_0, y_0)} f(x, y)$ exists.
$\lim_{(x,y) \to (x_0, y_0)} f(x, y) = f(x_0, y_0)$ .

If a function is continuous at every point in a region $R$ , it is continuous over $R$ . Polynomials, rational functions, and trigonometric functions are continuous on their domains.

1.4 Differentiability

A function $f(x, y)$ is differentiable at $(x_0, y_0)$ if its change $\Delta z$ can be expressed as:
$\Delta z = f_x(x_0, y_0)\Delta x + f_y(x_0, y_0)\Delta y + \varepsilon_1 \Delta x + \varepsilon_2 \Delta y$
where $\varepsilon_1, \varepsilon_2 \to 0$ as $(\Delta x, \Delta y) \to (0,0)$ .

Partial Derivatives: $f_x = \frac{\partial f}{\partial x}$ and $f_y = \frac{\partial f}{\partial y}$ must exist.
Total Differential: The total differential $dz$ (or $df$ ) is given by $dz = \frac{\partial f}{\partial x} dx + \frac{\partial f}{\partial y} dy$ .
Theorem: If the partial derivatives $f_x$ and $f_y$ exist and are continuous in a region containing $(x_0, y_0)$ , then $f(x, y)$ is differentiable at $(x_0, y_0)$ . (Differentiability implies continuity, but the mere existence of partial derivatives does not).

2. Chain Rule

The chain rule in multivariable calculus extends the single-variable concept to functions of multiple variables.

Case 1: One Independent Variable

If $z = f(x, y)$ is a differentiable function of $x$ and $y$ , where $x = x(t)$ and $y = y(t)$ are differentiable functions of $t$ , then $z$ is a function of $t$ , and:
$\frac{dz}{dt} = \frac{\partial f}{\partial x} \frac{dx}{dt} + \frac{\partial f}{\partial y} \frac{dy}{dt}$

Case 2: Two (or More) Independent Variables

If $z = f(x, y)$ , where $x = g(u, v)$ and $y = h(u, v)$ , then $z$ is a function of $u$ and $v$ . The partial derivatives are:
$\frac{\partial z}{\partial u} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial u} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial u}$
$\frac{\partial z}{\partial v} = \frac{\partial z}{\partial x} \frac{\partial x}{\partial v} + \frac{\partial z}{\partial y} \frac{\partial y}{\partial v}$

Tree Diagrams: Drawing a tree diagram (from $z$ to $x, y$ and then to $u, v$ ) helps map out the paths. You multiply derivatives along a path and add the results of different paths.

3. Change of Variables

Changing variables simplifies the integration or differentiation of complex functions, often moving from Cartesian coordinates to Polar, Cylindrical, or Spherical coordinates.

To change variables in partial derivatives, the chain rule is utilized. If changing from $(x, y)$ to $(u, v)$ , you solve for the new partials $\frac{\partial}{\partial u}$ and $\frac{\partial}{\partial v}$ in terms of the old ones.

In multiple integrals, changing variables requires the use of the Jacobian determinant to scale the area/volume element:
$dx dy = |J(u, v)| du dv$
(See section 5 for the detailed definition of the Jacobian).

4. Euler’s Theorem for Homogeneous Equations

4.1 Homogeneous Functions

A function $f(x, y)$ is said to be a homogeneous function of degree $n$ if, for any scalar $t > 0$ :
$f(tx, ty) = t^n f(x, y)$

4.2 Euler's Theorem

If $f(x, y)$ is a differentiable homogeneous function of degree $n$ , then:
$x \frac{\partial f}{\partial x} + y \frac{\partial f}{\partial y} = n f(x, y)$

4.3 Second-Order Euler's Theorem (Deduction)

By differentiating the first-order theorem, we obtain the second-order relation for a homogeneous function of degree $n$ :
$x^2 \frac{\partial^2 f}{\partial x^2} + 2xy \frac{\partial^2 f}{\partial x \partial y} + y^2 \frac{\partial^2 f}{\partial y^2} = n(n-1) f(x, y)$

5. Jacobians

5.1 Definition

If $u = u(x, y)$ and $v = v(x, y)$ are differentiable functions of $x$ and $y$ , the Jacobian of $u$ and $v$ with respect to $x$ and $y$ , denoted by $J$ or $\frac{\partial(u, v)}{\partial(x, y)}$ , is defined as the determinant of the Jacobian matrix:

$J = \frac{\partial(u, v)}{\partial(x, y)} = \begin{vmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{vmatrix} = \left(\frac{\partial u}{\partial x}\right)\left(\frac{\partial v}{\partial y}\right) - \left(\frac{\partial u}{\partial y}\right)\left(\frac{\partial v}{\partial x}\right)$

5.2 Properties of Jacobians

Chain Rule for Jacobians: If $u, v$ are functions of $x, y$ , and $x, y$ are functions of $r, s$ , then:
$\frac{\partial(u, v)}{\partial(r, s)} = \frac{\partial(u, v)}{\partial(x, y)} \cdot \frac{\partial(x, y)}{\partial(r, s)}$
Inverse Property: If $u, v$ are functions of $x, y$ , then:
$\frac{\partial(u, v)}{\partial(x, y)} \cdot \frac{\partial(x, y)}{\partial(u, v)} = 1 \implies J \cdot J' = 1$
Functional Dependence: If the Jacobian $\frac{\partial(u, v)}{\partial(x, y)} = 0$ identically in a region, then $u$ and $v$ are functionally dependent (i.e., there exists a relation $F(u, v) = 0$ ).

6. Extrema of Functions of Two Variables

Finding the maximum and minimum values (extrema) of surfaces in 3D space.

6.1 Critical Points

A point $(a, b)$ is a critical point of $f(x, y)$ if:

$f_x(a, b) = 0$ AND $f_y(a, b) = 0$ , OR
One or both partial derivatives do not exist at $(a, b)$ .

6.2 The Second Derivative Test

To classify the critical point $(a, b)$ where $f_x(a,b) = 0$ and $f_y(a,b) = 0$ , we use the discriminant (or Hessian determinant) $D$ :

$D = f_{xx}(a, b) \cdot f_{yy}(a, b) - [f_{xy}(a, b)]^2$

Let $r = f_{xx}(a,b)$ , $s = f_{xy}(a,b)$ , and $t = f_{yy}(a,b)$ . Then $D = rt - s^2$ .

Conditions:

If $D > 0$ and $f_{xx} > 0$ (or $r > 0$ ), then $f(a, b)$ is a Local Minimum.
If $D > 0$ and $f_{xx} < 0$ (or $r < 0$ ), then $f(a, b)$ is a Local Maximum.
If $D < 0$ , then $(a, b)$ is a Saddle Point (neither a max nor a min).
If $D = 0$ , the test is inconclusive (further analysis is needed).

7. Lagrange’s Method of Undetermined Multipliers

This method is used for constrained optimization: finding the local maxima and minima of a function subject to equality constraints.

7.1 The Setup

Objective Function: $f(x, y, z)$ (the function to maximize/minimize)
Constraint: $g(x, y, z) = c$ (or $g(x, y, z) - c = 0$ )

7.2 The Lagrangian Function

Introduce a new scalar variable $\lambda$ (the Lagrange multiplier) and construct the Lagrangian function $L$ :
$L(x, y, z, \lambda) = f(x, y, z) + \lambda [g(x, y, z) - c]$
(Note: Sometimes it is written as $-\lambda$ . Both yield the same optimal coordinates).

7.3 Solving the System

To find the critical points, take the partial derivatives of $L$ with respect to $x, y, z,$ and $\lambda$ , and set them to zero:

$\frac{\partial L}{\partial x} = \frac{\partial f}{\partial x} + \lambda \frac{\partial g}{\partial x} = 0$
$\frac{\partial L}{\partial y} = \frac{\partial f}{\partial y} + \lambda \frac{\partial g}{\partial y} = 0$
$\frac{\partial L}{\partial z} = \frac{\partial f}{\partial z} + \lambda \frac{\partial g}{\partial z} = 0$
$\frac{\partial L}{\partial \lambda} = g(x, y, z) - c = 0$ (which is just the constraint equation)

Solve this system of equations simultaneously to find the points $(x, y, z)$ . Evaluate the original function $f(x,y,z)$ at these points to determine the maximum and minimum values.

Unit 4

Unit 6