Understanding contours and level curves, drawing functions of several variables

This topic has me increasingly confused.

At some point I realised that essentially contours are really just a way of visualising 3d shapes in a 2d plane. Please correct me if I'm not entirely right.

for example, as seen here:

http://i.imgur.com/XgtalfW.jpg

But then during a lecture, the lecturer drew this:

http://i.imgur.com/MxyMzsC.jpg

Why are there dashed lines parallel to all sides of the triangle in the first example?

And the contour representation shown, wouldn't that form a rectangle and not a triangle? This has me incredibly confused.

We use the Stewarts Multivariable calculus (7th ed) textbook, would any of you recommend a different textbook or any online resources to understand this topic? I have a test on drawing functions of several variables in 4 days and just cannot seem to get a grasp on it, in addition to drawing their contours and level curves.

For the life of me I have no idea what the dashed lines in this diagram mean. Are they the contour lines for the graph of sin(x+y) ?

http://i.imgur.com/IhI2cJP.jpg

Re: Understanding contours and level curves, drawing functions of several variables

Hey 99.95.

The contours represent the region that lies on a particular plane.

The first image literally only has a cross section but the second one encompasses the whole plane since the triangle is itself a flat object on that plane that is defined by the relevant contour and a plane extends to an infinite area.

Re: Understanding contours and level curves, drawing functions of several variables

I'm starting to better understand this.

the MIT Lectures on this topic REALLY helped :)