Essential Concepts
- The graph of a function of two variables is a surface in R3R3 and can be studied using level curves and vertical traces.
- A set of level curves is called a contour map.
Key Equations
- Vertical trace
f(a,y)=xf(a,y)=x for x=ax=a or f(x,b)=zf(x,b)=z for y=by=b - Level surface of a function of three variables
f(x,y,z)=cf(x,y,z)=c
Glossary
- contour map
- a plot of the various level curves of a given function f(x,y)f(x,y)
- function of two variables
- a function z=f(x,y)z=f(x,y) that maps each ordered pair (x,y)(x,y) in a subset DD of R2R2 to a unique real number zz
- graph of a function of two variables
- a set of ordered triples (x,y,z)(x,y,z) that satisfies the equation z=f(x,y)z=f(x,y) plotted in three-dimensional Cartesian space
- level curve of a function of two variables
- the set of points satisfying the equation f(x,y)=cf(x,y)=c for some real number cc in the range of ff
- level surface of a function of three variables
- the set of points satisfying the equation f(x,y,z)=cf(x,y,z)=c for some real number cc in the range of ff
- surface
- the graph of a function of two variables, z=f(x,y)z=f(x,y)
- vertical trace
- the set of ordered triples (c,y,z)(c,y,z) that solves the equation f(c,y)=zf(c,y)=z for a given constant x=cx=c or the set of ordered triples (x,d,z)(x,d,z) that solves the equation f(x,d)=zf(x,d)=z for a given constant y=dy=d
Candela Citations
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- Calculus Volume 3. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: https://openstax.org/books/calculus-volume-3/pages/1-introduction. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at https://openstax.org/books/calculus-volume-3/pages/1-introduction