Chapter 11 - Surface Integrals

Question 1

Surface in R^3

Answer 1

The image S= F (D) = {F(u,v) such that (u,v) is an element of D}

Question 2

The edge of S

Answer 2

dS = F (c)

Question 3

The map F

Answer 3

A parameterisatiob of the surface

Question 4

Smooth surface

Answer 4

If the derivative matrix D(F)(u,v) has rank 2 (i.e. the two columns are linearly independent) for all points (u,v) in D

Question 5

Singular point

Answer 5

A point F(u,v) on the surface S is singular if the rank of D(F)(u,v) is less than 2

Question 6

Open surface

Answer 6

Every pair of points in R^3 not lying in S can be joined by a continuous curve that doesn’t cross S

Question 7

Closed surface

Answer 7

It divides R^3 into distinct regions, R1 and R2, such that every continuous curve joining a point in R1 to a point in R2 crosses S at least once

Question 8

Simple surface

Answer 8

The union of a finite number of smooth surfaces. Also referred to as piecewise smooth or regular

Question 9

The vector N

Answer 9

N = (dr/du) * (dr/dv). This is a non-zero vector that it normal to the surface s at point P. This occurs when (dr/du) and (dr/dv) are linearly independent

Question 10

Unit vector n hat

Answer 10

N hat = (1/(|N|)) * N. This is a unit normal vector to S at P

Question 11

Two orientations of S

Answer 11

The two optiona for choice of the sign of n hat. They are designated the positive and negative orientations

Question 12

Oriented surface

Answer 12

This occurs once a particular side of a surface has been chosen as positive

Question 13

Surface element

Answer 13

Let P0(u,v) be a point on the surface S with position vector r(u,v). Consider infinitesimal displacements from P0 to P1(u + du, v), P2 (u, v + dv) and P3 (u + du, v + dv), where the points are joined by the u and v coordinate lines. The portion of S that is bounded by P0P1P3P2 is called a sueface element

Question 14

Surface area of S

Answer 14

Double integral of [|(dr/du) cross product (dr/dv)|] dudv

Question 15

Cross product of (dr/du) and (dr/dv)

Answer 15

If (dr/du) = (a,b,c) and (dr/dv) = (d,e,f), then the cross product is equal to the determinant of the matrix
ijk
abc
deg

Question 16

Integral of (phi) dS

Answer 16

Double integral of [phi (u,v) * |cross product of (dr/du and dr/dv)| dudv

Question 17

Integral of v • dS

Answer 17

= integral of (v • n hat)dS

= double integral of [v (u,v) • (cross product of dr/du and dr/dv)] dudv

Question 18

Flux of v across S

Answer 18

Phi(v) = integral of [v (x,y,z) • n hat (x,y,z)dS

Question 19

Divergence theorem

Answer 19

Let R be a region of R^3 bounded by a closed simple surface S, which is oriented outwards.
If v (x,y,z) is a vector field whose components have continuous partial derivatives on some open set containing R, and if n hat (x,y,z) is the outward-directrd normal at position (x,y,z) on S, then 
Integral over S of (v•n hat)dS = integral over R of (div v) dV

Question 20

Stokes’ Theorem

Answer 20

Let S be a piecewise smooth orinted surface that is bounded by a simple, closed piecewise smooth curve C with positive orientation.
If v (x,y,z) is a continuous vector field with continuous first partial derivatives on some open set containing S, then
Closed integral over C of (v•dr) = integral over c of (curlv • n hat)dS
Where n hat is the cross product of dr/du and dr/dv