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# Introduction to the surface integral

Introduction to the surface integral. Created by Sal Khan.

## Want to join the conversation?

• what is the difference between surface integral and double integral ??
• There isn't one really. Taking a normal double integral is just taking a surface integral where your surface is some 2D area on the s-t plane. The general surface integrals allow you to map a rectangle on the s-t plane to some other crazy 2D shape (like a torus or sphere) and take the integral across that thing too!
• The magnitude of the cross product (of the two vector function partial derivatives) is nothing more than a Jacobean accounting for the distortion created when there was a change of variable right?
• Almost. In this case the Jacobian matrix is 3x2 and has no determinant, so first you need to left-multiply by the transpose to get a 2x2 matrix then take the determinant. But now you've basically done the transformation twice so you need to take the square root, and THEN you have the appropriate term to account for distortion (the "volume stretching factor" as they called it in my class).
• at he introduce the surface area to be a double integral, why? isn't the double integral the volume under the surface?
• The notations for the double integral and surface integral look the same for a good reason. In the double integral you're talking about, per every point in the region, you are adding a "column": volume given by f(x,y)dx dy. In the surface integral, per every point in the surface (region), you are adding a (parallelogram area times a scalar): represented by f(x, y, z) d-sigma. Thus in both cases the notations appropriately add up the expressions within the integral sign per point in the surface, but since the inner expressions are different, different things are calculated.
BTW: In the surface integral, the "region" is the surface itself.
tails to tails when you subtract them
example r1, r2 are vectors
to get r1-r2, place them tails to tails. the result is the vector pointing from head of r2 to head of r1
• So how can we use these concepts to derive the equation for the surface area of a sphere?
• What is the "formal" guarantee that we can equate a curved surface to a flat parallelogram as he says in ? Why does the fact that the area is "very very small" mean that that approximation is valid?
• Are the symbols for magnitude and absolute value the same?
(1 vote)
• Yes, they are. If you think about it, it makes sense. Since the magnitude of a vector is always non-negative, it makes sense to symbolize it the same way you symbolize absolute value.

Note however, magnitude is also symbolized with double bars,
as in || U ||
• can you use this to show that if h=f(s,t) then dh = delf/dels *ds +delf/delt *dt ?? If so, how?