4.1 Functions of Several Variables

The graph of a function of two variables is a surface in $ℝ^{3}$ and can be studied using level curves and vertical traces.
A set of level curves is called a contour map.

4.2 Limits and Continuity

To study limits and continuity for functions of two variables, we use a $δ$ disk centered around a given point.
A function of several variables has a limit if for any point in a $δ$ ball centered at a point $P,$ the value of the function at that point is arbitrarily close to a fixed value (the limit value).
The limit laws established for a function of one variable have natural extensions to functions of more than one variable.
A function of two variables is continuous at a point if the limit exists at that point, the function exists at that point, and the limit and function are equal at that point.

A partial derivative is a derivative involving a function of more than one independent variable.
To calculate a partial derivative with respect to a given variable, treat all the other variables as constants and use the usual differentiation rules.
Higher-order partial derivatives can be calculated in the same way as higher-order derivatives.

The analog of a tangent line to a curve is a tangent plane to a surface for functions of two variables.
Tangent planes can be used to approximate values of functions near known values.
A function is differentiable at a point if it is ”smooth” at that point (i.e., no corners or discontinuities exist at that point).
The total differential can be used to approximate the change in a function $z = f (x_{0}, y_{0})$ at the point $(x_{0}, y_{0})$ for given values of $Δ x$ and $Δ y .$

The chain rule for functions of more than one variable involves the partial derivatives with respect to all the independent variables.
Tree diagrams are useful for deriving formulas for the chain rule for functions of more than one variable, where each independent variable also depends on other variables.

A directional derivative represents a rate of change of a function in any given direction.
The gradient can be used in a formula to calculate the directional derivative.
The gradient indicates the direction of greatest change of a function of more than one variable.

A critical point of the function $f (x, y)$ is any point $(x_{0}, y_{0})$ where either $f_{x} (x_{0}, y_{0}) = f_{y} (x_{0}, y_{0}) = 0,$ or at least one of $f_{x} (x_{0}, y_{0})$ and $f_{y} (x_{0}, y_{0})$ do not exist.
A saddle point is a point $(x_{0}, y_{0})$ where $f_{x} (x_{0}, y_{0}) = f_{y} (x_{0}, y_{0}) = 0,$ but $(x_{0}, y_{0})$ is neither a maximum nor a minimum at that point.
To find extrema of functions of two variables, first find the critical points, then calculate the discriminant and apply the second derivative test.

An objective function combined with one or more constraints is an example of an optimization problem.
To solve optimization problems, we apply the method of Lagrange multipliers using a four-step problem-solving strategy.