### Key Concepts

### 4.1 Functions of Several Variables

- The graph of a function of two variables is a surface in ${\mathbb{R}}^{3}$ 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 $\delta $ disk centered around a given point.
- A function of several variables has a limit if for any point in a $\delta $ 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.

### 4.3 Partial Derivatives

- 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.

### 4.4 Tangent Planes and Linear Approximations

- 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 $z=f\left({x}_{0},{y}_{0}\right)$ at the point $\left({x}_{0},{y}_{0}\right)$ for given values of $\text{\Delta}x$ and $\text{\Delta}y.$

### 4.5 The Chain Rule

- 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.

### 4.6 Directional Derivatives and the Gradient

- 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.

### 4.7 Maxima/Minima Problems

- A critical point of the function $f\left(x,y\right)$ is any point $\left({x}_{0},{y}_{0}\right)$ where either ${f}_{x}\left({x}_{0},{y}_{0}\right)={f}_{y}\left({x}_{0},{y}_{0}\right)=0,$ or at least one of ${f}_{x}\left({x}_{0},{y}_{0}\right)$ and ${f}_{y}\left({x}_{0},{y}_{0}\right)$ do not exist.
- A saddle point is a point $\left({x}_{0},{y}_{0}\right)$ where ${f}_{x}\left({x}_{0},{y}_{0}\right)={f}_{y}\left({x}_{0},{y}_{0}\right)=0,$ but $\left({x}_{0},{y}_{0}\right)$ 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.

### 4.8 Lagrange Multipliers

- 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.