Level sets and contour maps, Multivariable Calculus

This lecture is about level sets for functions z=f(x,y) or w=f(x,y,z). We sketch contour diagrams consisting of several level sets. The difference between level sets and cross sections. If you are visiting my channel, please subscribe and check out the rest of my videos on Multivariable Calculus!

Multivariable Calculus Unit 3 Lecture 3

Level Curves for f(x,y)=x^2+y^2: We consider the function f(x,y)=x^2+y^2 and determine its level curves. For a fixed C, the level curve is defined by the equation x^2+y^2=C. We note that:
If C is negative, no points satisfy this.
If C=0, only the origin satisfies this, forming the zero level set.
If C is positive, the level curve is a circle with radius sqrt(C).

We sketch a contour diagram for various values of C, resulting in concentric circles in the xy-plane.

Level Curves for f(x,y)=cos⁡(xy): Next, we find level curves for f(x,y)=cos⁡(xy). For a constant C, the equation becomes cos⁡(xy)=C. We observe:
If C=1, the level curve includes the axes x=0 or y=0.
For other values of C, the level curves form hyperbolas defined by xy=k+2πn, where cos⁡(k)=C.

A contour diagram of this function shows hyperbolic level curves, helping us visualize the function's shape.

Similarly, we look at level surfaces. Then we clarify the difference between cross sections and level sets for function z=f(x,y):
Cross sections are part of the graph of the function and are drawn in R^3 (for functions of two variables).
Level sets belong to the domain of the function and are drawn in R2 (for functions of two variables).

For example, the level sets of z=x^2+y^2 are concentric circles in the xy-plane, while a cross section like y=0 is a curve on the graph of the function in R3.

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