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### Course: Multivariable calculus>Unit 4

Lesson 10: Surface integral preliminaries

# Determining a position vector-valued function for a parametrization of two parameters

Determining a Position Vector-Valued Function for a Parametrization of Two Parameters. Created by Sal Khan.

## Want to join the conversation?

• I really dont get why b+acos(s) is a hypotenus? Watching the video i understood it like b was a fixed distance from z on the y-plane? And that s is an angle from the y-plane. So for me b+ acos(s) seems to be y(s,t)....but apparrently is not? Please help:)
• It might help to think of b + a*cos(s) as the distance from the origin of the xy-plane. If so, you would need to break it into its x- and y-components by multiplying this distance by sin(t) and cos(t), respectively.
• I found it very confusing the changing of sign on the x-axis. We should keep the right-hand orientation system.
• Right. This choice of X direction makes the cross product of i and j equal -k

i x j = -k

This is a no-no in physics.
• Wouldn't it be more natural to define the coordinates from a right-hand rule coordinate system?
If I do that then I get these coordinates:

x = [b + cos(s)] · cos(t)
y = [b + sin(s)] · sin(t)
z = b · sin(s)

I think it looks nicer if the x- and y-axes have cosine and sine values respectively like that.
• Yes, you could have oriented the coordinate system like that if you wanted.
• Amazing tutorial! are there videos of other surfaces apart from the taurus?
• it would be really nice if there were some examples that werent the taurus
• Shouldn't the unit vector i go into the other direction of the x-axis? (Minute ) Otherwise we set up a left-handed coordinate system, didn't we?
(1 vote)
• Yes, Sal decided to change the sign of the x coordinate on minute , turning the system into a left-handed coordinate system.
• This is the first time I have heard about a left-hand coordinate system. Are there situations when it would be an advantage to use such a system? It just seems confusing to me since I am not used to it.
• Why the subtitles are not visible in all the videos these days?
• What would be other notable examples of surfaces defined by two parameters?
(1 vote)
• Spheres, open cylinders and ellipsoids.
A more general example would be any surface defined by a scalar function z = f(x,y). The parametrization here would be:

x(s,t) = s
y(s,t) = t
z(s,t) = f(s,t)

This allows you to parametrize paraboloid, hyperboloids, etc.

Then there are 'surfaces' that can be continously described with two parameteres and a distance function but cannot be realized in three dimensional Euclidean space. So you go from:

A surface is a two dimensional segment of the three dimensional space.

to

A surface is whatever that can be described by two parameters and an appropriate rule for distance.