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# Surface integral example part 2

Taking the cross product to calculate the surface differential in terms of the parameters. Created by Sal Khan.

## Want to join the conversation?

• Where can I find Sal's videos on determinants and cross-products, if any? Thanks!
• In the Linear Algebra playlist, click Vectors and Spaces and scroll down.
• Wait, the title of this video says it's a differential? I thought it would be an integral. What does differential even mean? Thanks
• The differential has more in common with a derivative than an integral. However, it is not exactly the same as a derivative. They are both rates of change, but the derivative of a function is the rate of change of the function with respect to its independent variable(s) (e.g. dy/dx, df/dx, df/dy, etc.). The differential of the function, on the other hand, is the rate of change of the function as a whole, without respect to its variables (e.g. df, dy, dz, etc.)
• Why is the paramaterization cos(t)cos(s)i+sin(t)sin(s)j+sin(t)k?
Shouldn't it be rsin(t)cos(s)i+rsin(t)sin(s)j+rsin(t)k?
• At the end shouldn't it be sqrt(cos^4(t) + 1) not cos(t)?
• The t's and the +'s look very similar in the video, I think you meant that you end up with sqrt(cos^4(t)+cos^2(t)sin^2(t)), not sqrt(cos^4(t)+sin^2(t)+cos^2(t). Then you factor out a cos^2(t) and you get sqrt(cos^2(t) * (cos^2(t) + sin^2(t))) which simplifies to
sqrt(cos^2(t) * 1 which in turn becomes cos(t). Hope this helped
• you could've just told about jacobian which is more general than cross product
• in about . why the order of the cross product is Rs *Rt. But not the reversed order? I dont know how to tell which order to use? Thank you very much.
• The order doesn't matter because either way the magnitude will be the same.
Essentially Rs*Rt = -Rt*Rs, if you reverse the order you get a minus sign, but when you take the magnitude the minus sign is irrelevant.
• Why is cos t the magnitude of the cross product? Clearly, it can take on negative values.
(1 vote)
• The way sphere parameterization was defined it actually can't, since t only goes from -pi/2 to pi/2.
• you are saying a verbal misstake between 0.52-0.57 by saying the crossproduct of the parameterization with respect to the parameter whilst you really meant to say the partial with respect to the parametrization as you did in the first case just a couple of seconds before that
(1 vote)
• at 12.29 sal has treated the second (sin^2+cos^2=1)as a coefficient, should it not be cos^4(t) +1?
(1 vote)
• sorry my bad, forget that
(1 vote)
• At when he does the square root, can you take out the cos(t) and use it as a scalar? He did something similar to that with the torus example. Does leaving it in just make it simpler to, well simplify.
It would look like:
Cos(t) * sqrt(magnitude)
(1 vote)