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**Vector Calculus: Surface Integrals**

## Homework Statement

Find the surface integral of

**u[/B dot**

How do you "find" the surface. I have just started ont the subject and I have no idea how to see what is the surface and the region of integration.

Solution according to the book

The surface is written parametically as (x,y,x+y^2)

two vectors parallel to the surface are (1,0,1) and (0,1,2y)

Their cross product = (-1,-2y,1)

changing the direction of

region of integration x+y^2 < 0, x > -1, so doing the x integration first, -1<x<-y^2 and

-1 <y <1

**n**over S where S is part of the surface z = x + y^2 with z < 0 and x > -1,**u**is the vector field u = (2y,x -1,0) and**n**has a negative z component## Homework Equations

## The Attempt at a Solution

How do you "find" the surface. I have just started ont the subject and I have no idea how to see what is the surface and the region of integration.

Solution according to the book

The surface is written parametically as (x,y,x+y^2)

two vectors parallel to the surface are (1,0,1) and (0,1,2y)

Their cross product = (-1,-2y,1)

**n**dS = (-1,-2y,1)dxdychanging the direction of

**n****n**dS = (1,2y,-1)dxdy**u**dot**n**dS = xdxdyregion of integration x+y^2 < 0, x > -1, so doing the x integration first, -1<x<-y^2 and

-1 <y <1

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