Question

Explain the Surface Area equation

Answer 1

Okay so can you type the actual equation?

Answer 2

where T is any region

Answer 3

in x y plane

Answer 4

Okay do you understand the equation for a parametrized curve?
Such as \(r(u,v) = (x(u,v),y(u,v),z(u,v))\)?

Answer 5

kind offff Not really

Answer 6

Okay, then what do you understand so far?

Answer 7

i know the equation u said
just x and y and z are a fuction of u and v and r is the position vector

Answer 8

i know the surface integral is something from the formula where u cross 2 vectors and the magnitude of it is the area

Answer 9

so is its u and v it wud be me magnitude of U X V

Answer 10

which i dont get why

Answer 11

You know that for a curve \(\mathbf{r}(u,v)= (x(u,v),y(u,v),z(u,v))\) Where \((u,v)\in D\) the surface area area is \[
\iint_Sd\mathbf{S} = \iint_D||\mathbf{r}_u\times\mathbf{r}_v||\; dA
\]

Answer 12

This equation does not cause any confusion, right?

Answer 13

i know the formula thats about it but i dont know anything about why that equation is true

Answer 14

Well in this case, what they're saying is that if the parametrization is \(\mathbf{r}(x,y) = (x, y, f(x,y))\) then you have \(\mathbf{r}_x = (1, 0, f_x)\) and \(\mathbf{r}_y = (0, 1, f_y)\)

Answer 15

This sort of parametrization is a graph parametrization, and it's nice because we can quickly find the magnitude of the normal vector. We know it will be \(\sqrt{f_x^2+f_y^2+1}\)

Answer 16

is that form the cross of rx * ry

Answer 17

llRx X Ryll

Answer 18

Well \(\mathbf{r}_x\times \mathbf{r}_y = (f_x, f_y, -1)\). It's the magnitude of it.

Answer 19

ya i see that but why does that give your area of the surface

Answer 20

Well, remember how we did arc length?

Answer 21

thats the part iam confused about they said something this being from that part, but i forgot how we did arc length for y = f(x)

Answer 22

from equation s =