Multivariable Calculus: Functions, Limits, Derivatives, and Optimization

Multivariable Functions and Their Domains

Definition and Domain of Multivariable Functions

Multivariable functions extend the concept of functions to more than one input variable, commonly written as f(x, y) or f(x, y, z). The domain of a multivariable function is the set of all input values for which the function is defined.

• Key Point: To find the domain, identify all restrictions (such as division by zero or taking the logarithm of a negative number).

• Example: For $f(x, y) = \frac{x^2 - y}{xy}$, the domain excludes points where $xy = 0$.

Open and Closed Sets

A set in the plane or space is open if it does not include its boundary points, and closed if it includes all its boundary points. Some sets can be neither or both.

• Key Point: To determine openness or closedness, examine whether boundary points are included.

• Example: The set $\{(x, y) \mid x^2 + y^2 < 1\}$ is open; $\{(x, y) \mid x^2 + y^2 \leq 1\}$ is closed.

Level Curves

Level curves (contour lines) of a function $f(x, y)$ are curves along which the function takes a constant value. They are useful for visualizing functions of two variables.

• Key Point: To sketch level curves, set $f(x, y) = c$ for various constants $c$ and solve for $x$ and $y$.

• Example: For $f(x, y) = 4x - 4y + 8$, level curves are lines $4x - 4y + 8 = c$.

Limits and Continuity in Multivariable Functions

Evaluating Multivariable Limits

Limits in multivariable calculus involve approaching a point from infinitely many directions. A limit exists only if the function approaches the same value from all paths.

• Plug and Chug: Substitute the point directly if the function is continuous at that point.

• Simplification: Simplify the expression before substitution to avoid indeterminate forms.

• Path Analysis: To show a limit does not exist, find two different paths approaching the point that yield different limit values.

• Example: $\lim_{(x, y) \to (0, 0)} \frac{x^2 - y^2}{xy - 1}$

Partial Derivatives and the Multivariable Chain Rule

Partial Derivatives

A partial derivative measures the rate of change of a function with respect to one variable, holding others constant. Notations include $\frac{\partial f}{\partial x}$, $f_x$, and $\partial_x f$.

• Second Order Partial Derivatives: These are derivatives of partial derivatives, such as $\frac{\partial^2 f}{\partial x^2}$ or $\frac{\partial^2 f}{\partial x \partial y}$.

• Example: For $f(x, y, z) = 5x \cos(xy) - e^z \ln(z^3)$, compute $f_{xx}$, $f_{xy}$, $f_{xz}$, etc.

Approximating Derivatives Using Tables

When a function is given as a table of values, derivatives can be approximated using finite differences.

• Key Point: Use $\frac{f(x+h, y) - f(x, y)}{h}$ for small $h$ to approximate $f_x$.

Multivariable Chain Rule

The chain rule in multivariable calculus allows differentiation of composite functions. If $f(x, y) = g(u(x, y), v(x, y))$, then:

• Formula: $f_x = g_u \cdot u_x + g_v \cdot v_x$

• Example: Given values for $g_u$, $g_v$, $u_x$, $v_x$ at a point, substitute to find $f_x$.

Gradient and Directional Derivatives

Gradient Vector

The gradient of a function $f(x, y)$ is the vector of partial derivatives: $\nabla f = \langle f_x, f_y \rangle$. It points in the direction of greatest increase of the function.

• Significance: The gradient is orthogonal to level curves and surfaces.

• Example: For $f(x, y) = x^2y - e^{xy}$, $\nabla f$ at $(1, 2)$ is $\langle f_x(1, 2), f_y(1, 2) \rangle$.

Directional Derivative

The directional derivative

• Formula: $D_{\mathbf{u}} f = \nabla f \cdot \mathbf{u}$, where $\mathbf{u}$ is a unit vector.

• Example: Find the rate of change at $(1, 2)$ in the direction $\langle 5, -1 \rangle$.

Tangent Planes and Linearization

Tangent Plane to a Surface

The tangent plane

• Formula: $z = f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)$

• Example: For $f(x, y) = x^2y - e^{xy}$ at $(1, 2)$, compute $f_x$ and $f_y$ at $(1, 2)$ and substitute.

Tangent Plane to a Level Surface

For a level surface $F(x, y, z) = c$, the tangent plane at $(x_0, y_0, z_0)$ is:

• Formula: $F_x(x_0, y_0, z_0)(x - x_0) + F_y(x_0, y_0, z_0)(y - y_0) + F_z(x_0, y_0, z_0)(z - z_0) = 0$

• Example: For $xyz - xz + 3xy = 5$ at $(1, 2, -1)$, compute partials and substitute.

Tangent Line to Intersection of Surfaces

The tangent line to the intersection of two surfaces can be found by taking the cross product of their gradients at the point of intersection.

• Key Point: Compute $\nabla F$ and $\nabla G$ at the point, then $\mathbf{v} = \nabla F \times \nabla G$ gives the direction vector.

Optimization: Critical Points and Constrained Optimization

Critical Points and Classification

Critical points of a function are points where all partial derivatives are zero or undefined. They are candidates for local minima, maxima, or saddle points.

• Classification: Use the second derivative test for functions of two variables:

• Formula: $D = f_{xx}f_{yy} - (f_{xy})^2$

• If $D > 0$ and $f_{xx} > 0$, local minimum; if $D > 0$ and $f_{xx} < 0$, local maximum; if $D < 0$, saddle point.

• Example: For $f(x, y) = x^3 + y^3 + 3x^2 - 3y^2 - 8$, find critical points and classify.

Setting Up Optimization Problems

Optimization problems involve finding the maximum or minimum value of a function, sometimes subject to constraints.

• Key Point: For unconstrained optimization, set partial derivatives to zero and solve.

• Constrained Optimization: Use the method of Lagrange multipliers for constraints.

• Example: Find the largest product of positive numbers $x, y, z$ such that $x + y + z^2 = 16$.

Global Min/Max with Constraints

To find global extrema subject to a constraint, set up the Lagrangian:

• Formula: $L(x, y, \lambda) = f(x, y) - \lambda (g(x, y) - c)$

• Solve $\nabla L = 0$ for $x, y, \lambda$.

Summary Table: Classification of Critical Points

Condition	Classification
$D > 0$, $f_{xx} > 0$	Local Minimum
$D > 0$, $f_{xx} < 0$	Local Maximum
$D < 0$	Saddle Point
$D = 0$	Test Inconclusive

Additional info: These notes cover topics from multivariable calculus, typically found in chapters beyond single-variable calculus, such as functions of several variables, partial derivatives, gradients, tangent planes, and optimization. The content is relevant for students studying Calculus III or advanced calculus courses.