Essential Concepts
Functions of Several Variables
- The graph of a function of two variables is a surface in [latex]\mathbb{R}^{3}[/latex] and can be studied using level curves and vertical traces.
- A set of level curves is called a contour map.
Limits and Continuity
- To study limits and continuity for functions of two variables, we use a [latex]\delta[/latex] disk centered around a given point.
- A function of several variables has a limit if for any point in a [latex]\delta[/latex] ball centered at a point [latex]P[/latex], 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.
Introduction to 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.
Key Equations
- Vertical trace
[latex]f(a,y)=x[/latex] for [latex]x=a[/latex] or [latex]f(x,b)=z[/latex] for [latex]y=b[/latex] - Level surface of a function of three variables
[latex]f(x,y,z)=c[/latex] - Partial derivative of [latex]f[/latex] with respect to [latex]x[/latex]
[latex]\frac{\partial f}{\partial x}=\underset{h \to{0}}{\lim} \frac{f(x+h,y)-f(x,y)}{h}[/latex] - Partial derivative of [latex]f[/latex] with respect to [latex]y[/latex]
[latex]\frac{\partial f}{\partial y}=\underset{k \to{0}}{\lim} \frac{f(x,y+k)-f(x,y)}{k}[/latex]
Glossary
- boundary point
- a point [latex]P_{0}[/latex] of [latex]R[/latex] is a boundary point if every [latex]\delta[/latex] disk centered around [latex]P_{0}[/latex] contains points both inside and outside [latex]R[/latex]
- closed set
- a set [latex]S[/latex] that contains all its boundary points
- connected set
- an open set [latex]S[/latex] that cannot be represented as the union of two or more disjoint, nonempty open subsets
- contour map
- a plot of the various level curves of a given function [latex]f(x,y)[/latex]
- function of two variables
- a function [latex]z=f(x,y)[/latex] that maps each ordered pair [latex](x,y)[/latex] in a subset [latex]D[/latex] of [latex]\mathbb{R}^{2}[/latex] to a unique real number [latex]z[/latex]
- graph of a function of two variables
- a set of ordered triples [latex](x,y,z)[/latex] that satisfies the equation [latex]z=f(x,y)[/latex] plotted in three-dimensional Cartesian space
- higher-order partial derivatives
- second-order or higher partial derivatives, regardless of whether they are mixed partial derivatives
- interior point
- a point [latex]P_{0}[/latex] of [latex]R[/latex] is a boundary point if there is a [latex]\delta[/latex] disk centered around [latex]P_{0}[/latex] contained completely in [latex]R[/latex]
- level curve of a function of two variables
- the set of points satisfying the equation [latex]f(x,y)=c[/latex] for some real number [latex]c[/latex] in the range of [latex]f[/latex]
- level surface of a function of three variables
- the set of points satisfying the equation [latex]f(x,y,z)=c[/latex] for some real number [latex]c[/latex] in the range of [latex]f[/latex]
- mixed partial derivatives
- second-order or higher partial derivatives, in which at least two of the differentiations are with respect to different variables
- open set
- a set [latex]S[/latex] that contains none of its boundary points
- partial derivative
- a derivative of a function of more than one independent variable in which all the variables but one are held constant
- partial differential equation
- an equation that involves an unknown function of more than one independent variable and one or more of its partial derivatives
- region
- an open, connected, nonempty subset of [latex]\mathbb{R}^{2}[/latex]
- surface
- the graph of a function of two variables, [latex]z=f(x,y)[/latex]
- vertical trace
- the set of ordered triples [latex](c,y,z)[/latex] that solves the equation [latex]f(c,y)=z[/latex] for a given constant [latex]x=c[/latex] or the set of ordered triples [latex](x,d,z)[/latex] that solves the equation [latex]f(x,d)=z[/latex] for a given constant [latex]y=d[/latex]
- [latex]\delta[/latex] ball
- all points in [latex]\mathbb{R}^{3}[/latex] lying at a distance of less than [latex]\delta[/latex] from [latex](x_{0},y_{0},z_{0})[/latex]
- [latex]\delta[/latex] disk
- an open disk of radius [latex]\delta[/latex] centered at point [latex](a,b)[/latex]