How do you sketch level curves of multivariable functions? Vector Calculus #3

In this video we’re talking about how to sketch the level curves of a multivariable function.

Whenever you’re dealing with a multivariable function, the graph of that function will be a three-dimensional figure in space. If you take a perfectly horizontal sheet or plane that’s parallel to the xy-plane, and you use that to slice through your three-dimensional figure, then what you get at the intersection of the figure and the plane is a two-dimensional curve.

What we want to be able to do is slice through the figure at all different heights in order to get what we call the “level curves” of a function. Then we want to be able to transfer all those two-dimensional curves into the two-dimensional plane, sketching those in the xy-plane.

This will give us the sketch of level curves of the function. In this video we’re going to talk about how to find the level curves both graphically (by looking at a picture of the three-dimensional figure) and algebraically, by replacing z in the multivariable function with a constant c, and then substituting different values for c in order to get equations that are in terms of x and y only and can therefore be graphed in the xy-plane.


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