| | 2025.01.28

4: Differential Calculus of Several Variables

Extending derivatives to functions of multiple variables.

Functions of Several Variables

A function $f: \mathbb{R}^n \to \mathbb{R}$ assigns a real number to each point in $n$ -dimensional space.

Examples:

$f(x, y) = x^2 + y^2$ (paraboloid)
$f(x, y) = \sqrt{1 - x^2 - y^2}$ (hemisphere)
$f(x, y, z) = x^2 + y^2 + z^2$ (distance squared from origin)

Level Curves and Surfaces

Level curve: Set of points where $f(x, y) = c$ (constant)

Level surface: Set of points where $f(x, y, z) = c$

These are the “contour lines” on a topographic map.

Limits and Continuity

\lim_{(x,y) \to (a,b)} f(x, y) = L

means $f(x, y)$ approaches $L$ as $(x, y)$ approaches $(a, b)$ along any path.

Showing a Limit Doesn’t Exist

Find two different paths to $(a, b)$ that give different limits.

Example: For $f(x, y) = \frac{xy}{x^2 + y^2}$ :

Along $y = 0$ : $\lim = 0$
Along $y = x$ : $\lim = \frac{1}{2}$

So the limit doesn’t exist at $(0, 0)$ .

Partial Derivatives

Hold all variables constant except one, then differentiate:

\frac{\partial f}{\partial x} = f_x = \lim_{h \to 0} \frac{f(x+h, y) - f(x, y)}{h}

\frac{\partial f}{\partial y} = f_y = \lim_{h \to 0} \frac{f(x, y+h) - f(x, y)}{h}

Higher-Order Partials

f_{xx} = \frac{\partial^2 f}{\partial x^2}, \quad f_{yy} = \frac{\partial^2 f}{\partial y^2}, \quad f_{xy} = \frac{\partial^2 f}{\partial y \partial x}

Clairaut’s Theorem: If $f_{xy}$ and $f_{yx}$ are continuous, then $f_{xy} = f_{yx}$ .

The Gradient

The gradient of $f$ is the vector of partial derivatives:

\nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right\rangle

Key Properties

$\nabla f$ points in the direction of steepest increase
$|\nabla f|$ is the rate of steepest increase
$\nabla f$ is perpendicular to level curves/surfaces

Directional Derivatives

The rate of change of $f$ in direction $\mathbf{u}$ (unit vector):

D_\mathbf{u} f = \nabla f \cdot \mathbf{u} = |\nabla f| \cos\theta

Maximum: $D_\mathbf{u} f = |\nabla f|$ when $\mathbf{u}$ is parallel to $\nabla f$

Minimum: $D_\mathbf{u} f = -|\nabla f|$ when $\mathbf{u}$ is opposite to $\nabla f$

Zero: $D_\mathbf{u} f = 0$ when $\mathbf{u} \perp \nabla f$ (along level curve)

Chain Rule

Case 1: $z = f(x, y)$ where $x = g(t)$ , $y = h(t)$

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

Case 2: $z = f(x, y)$ where $x = g(s, t)$ , $y = h(s, t)$

\frac{\partial z}{\partial s} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial s} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial s}

\frac{\partial z}{\partial t} = \frac{\partial f}{\partial x}\frac{\partial x}{\partial t} + \frac{\partial f}{\partial y}\frac{\partial y}{\partial t}

Implicit Differentiation

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

\frac{dy}{dx} = -\frac{F_x}{F_y}

Tangent Planes and Linear Approximation

Tangent Plane

To surface $z = f(x, y)$ at $(a, b, f(a, b))$ :

z - f(a, b) = f_x(a, b)(x - a) + f_y(a, b)(y - b)

Linear Approximation

f(x, y) \approx f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b)

Total Differential

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

Optimization

Critical Points

Points where $\nabla f = \mathbf{0}$ (or gradient doesn’t exist).

f_x = 0 \quad \text{and} \quad f_y = 0

Second Derivative Test

At critical point $(a, b)$ , compute:

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

Condition	Conclusion
$D > 0$ and $f_{xx} > 0$	Local minimum
$D > 0$ and $f_{xx} < 0$	Local maximum
$D < 0$	Saddle point
$D = 0$	Test inconclusive

Lagrange Multipliers

To optimize $f(x, y)$ subject to constraint $g(x, y) = c$ :

Solve the system:

\nabla f = \lambda \nabla g

g(x, y) = c

Geometric interpretation: At the optimum, $\nabla f$ is parallel to $\nabla g$ (level curve of $f$ is tangent to constraint curve).

Summary

Concept	Formula
Partial derivative	$f_x = \lim_{h \to 0} \frac{f(x+h, y) - f(x,y)}{h}$
Gradient	$\nabla f = \langle f_x, f_y, f_z \rangle$
Directional derivative	$D_\mathbf{u} f = \nabla f \cdot \mathbf{u}$
Tangent plane	$z = f(a,b) + f_x(x-a) + f_y(y-b)$
Critical points	$\nabla f = \mathbf{0}$
Lagrange multipliers	$\nabla f = \lambda \nabla g$