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# Introduction to the surface integral

Introduction to the surface integral. Created by Sal Khan.

## Want to join the conversation?

• At or there about You introduce the concept of each point on the surface having a f(x,y,z) value.
Firstly what does evaluating this integral with f(x,y,z) represent? A volume between the two surfaces, or just the surface area of the new surface f(x,y,z) which takes value from the param. surface in order to 'create it self'?
If the second one (which I think is more likely) why not create a param. to map straight to this surface?
Or is it something in 4d?
• It's not a volume since f(x,y,z) is some scalar value defined for ANY point in 3D space. It could be an electric field strength function for example.

What Sal is saying at the end is that when you integrate "over the surface" which is the parameterized r(s,t) with the scalar function f(x,y,z) as part of the integrand, then you get a new value (and that can be useful in certain physics problems).

f(x,y,z) is not a surface itself - we are just "sampling" or "reading out" it's value at each point on the 2D surface defined by r(s,t). The last surface integral he gives is just a sum total of f over that whole weird surface shape (sigma).

In a way you can think of it in 4D since f(x,y,z) is a function of 3 variables and gives as output a 4th value.

All-in-all, I think this idea may be a bit confusing at first since it's very generic, but with some examples, especially ones in physics, then it'll be even more intuitive.
• 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).
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?
• 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.
• 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?