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### Course: Multivariable calculus>Unit 4

Lesson 10: Surface integral preliminaries

# Parametrizing a surface, part 1

Introduction to Parametrizing a Surface with Two Parameters. Created by Sal Khan.

## Want to join the conversation?

• I think it should be "3Pi/2" instead of "3Pi/4" at and .
• Yes. Sal made a mistake there. 3Pi/2 corresponds to 270 deg whereas 3Pi/4 corresponds to 135deg. But if you are able to correctly visualize the torus with respect to its orientation around the three axis, it shouldn't be a problem. :)
• Is there any program or website to help me visualize a torus in 3d space?
• Should the axes be labeled differently?

My understanding is that the bottom right (where Sal labeled the positive y-axis) should be the positive x-axis and the positive y-axis should be behind it. Because angles in standard position go counter-clockwise from the positive x-axis, I would start the rotations of t from the positive x-axis. I would like the origin on his s,t plane to begin on the positive x-axis, rather than the positive y-axis.
Looking at it this way, Sal's shadings of the torus at about make more sense to me.

Please help me to understand what I may have missed.
• Notice that the axes you described are the same as the axes Sal drew, just rotated a bit. The real important thing is that your space is what is called a right handed coordinate system. In such a space, x × y = z (that is the cross product).
As for the bit about the angles, all that matters is that the angle goes around counter-clockwise. If you wrap your fingers in the direction of increasing angle, your thumb should point in the direction of the +z axis.
• At Sal says S parameter describes the angle between radius and x-z plane. Has he mistaken it for x-y plane, considering the way he uses the parameter afterwards?
• Yes, he meant to say x-y plane. A little box pops up in the lower right corner of the screen with the correction.
• There were basic numerical mistakes Pi 3Pi over 2 (it was stated as 3Pi over 4) why wasn't it corrected
• Is it just me or do the areas he shades in at not match? He is right in that from 0 - pi/2 for S and T only sketches 1/4th of the outer surface for 1/4th of the circumference, or 1/8th of the total torus surface. But on the right diagram he starts shading this parametrization on the y-axis while the left he starts from the x-axis. It is important to maintain the same starting point so parameters and bounds align correctly
• Good catch, based on how he defined t, the diagram on the right is correctly shaded, but the upper left (top view) is not ...

I've put your observation into the Tips&Feedback, I think he is more likely to see those!
• isn't a doughnut a solid shape? how can we say it's a surface (2 D) ?
(1 vote)
• It is a surface in three-dimensional space, just like a sphere is a surface.
• Can someone please explain what the 's' parameter is for in this video? I don't understand.
(1 vote)
• I find it helpful to start by thinking of a more familiar circle drawn in 2 dimensions on an x-y coordinate system. This circle can be described with a radius, and the radius rotates through 2pi radians. If we call the radius of the circle 'r', and the angle it rotates through 's', we can parameterize this circle using x = r*cos(s) and y=r*sin(s).

Making the jump to 3 dimensions and describing a torus, you can think of it as a system of two circles. One of the circles describes the ring shape of the torus. The other circle describes the cross-section (if you took a very thin slice of one part of the torus). In the video, the 's' parameter is used to describe the cross section circle, and the 't' parameter is used to describe the overall ring shape of the torus.