Calc III - Graphing a Function of Multiple Variables by hand

[Cloudless]

I come across questions where I have to match the equation with its contour map and graph.

Examples:

z = sin(xy)

z = sin(x-y)


Right now I'm using Wolfram Alpha for all of these, but supposing these appear on an exam... how do I graph it by hand? For example, for z = sin(xy) I'm just plugging in random values for x and y... but doesn't that mean the entire xy graph should be filled then?

 

[kai_sikorski]

A contour of (for example) sin(xy) is a curve on the x, y plane where this function is constant. This in turn means that the product xy must be constant. So you could write

c = xy
y = c/x

And for each c you get a different contour. Plotting the curves once you have them in that form is probably a little more familiar to you.

 

[SammyS]

I come across questions where I have to match the equation with its contour map and graph.

Examples:

z = sin(xy)

z = sin(x-y)


Right now I'm using Wolfram Alpha for all of these, but supposing these appear on an exam... how do I graph it by hand? For example, for z = sin(xy) I'm just plugging in random values for x and y... but doesn't that mean the entire xy graph should be filled then?

Set z equal to some value (-1 ≤ z ≤ 1, why?), then solve for y in terms of x.

 

[kai_sikorski]

If you're doing it by hand I think it's easier to do what I wrote, for a few simple c values and afterwards evaluate what z is on each curve. Of course your way works too.

 

[Cloudless]

A contour of (for example) sin(xy) is a curve on the x, y plane where this function is constant. This in turn means that the product xy must be constant. So you could write

c = xy
y = c/x

And for each c you get a different contour. Plotting the curves once you have them in that form is probably a little more familiar to you.

Wow, this is brilliant @_@ Thank you

Set z equal to some value (-1 ≤ z ≤ 1, why?), then solve for y in terms of x.

You would need a calculator for this method though, right? Inverse sin of (-1 to 1) = xy.