CSV

Question 0

What are the cylindrical polar co-ordinates?

Answer 0

(p, phi, z) 
x=pcos(phi)
y=psin(phi)
Outward pointing normal of:
Curved surface: n=ep; Base: n=-ez;
Top: n=ez;
Position vector: r=pep+zez

Question 1

What is the dot product, cross product, perpendicular distance from origin to plane and the equation of a plane?

Answer 1

Dot product: a•b=|a||b|cos(theta);
Cross product: axb=|a||b|sin(theta)*n;
Perpendicular distance: r0•n;
Equation of plane: (r-r0)•n=0 (used to find the equation of the tangent plane to the surface)

Question 2

What are the spherical polar co-ordinates?

Answer 2

(r, theta, phi)
x=rsin(theta)cos(phi)
y=rsin(theta)sin(phi)
z=rcos(theta)
Outward pointing normal is n=er 
The position vector: r=rer

Question 3

What is grad f ?

Answer 3

Df/Dx i + Df/Dy j + Df/Dz k
Cylindrical: 
Df/Dp ep + 1/p Df/D(phi) ephi + Df/Dz ez
Spherical:
Df/Dr er + 1/r Df/D(theta) etheta + 1/rsin(theta) Df/Dphi ephi

Question 4

What is the directional derivative?

Answer 4

In the direction, say u, it is:
Duf(a,b,c) = u’•gradf(a,b,c), where
u’= u/|u| is the unit vector in the direction of u.

Question 5

What is Schwarz’s Theorem?

Answer 5

Suppose that z=f(x,y) is continuous and fx, fy, fxy, fyx are all continuous then D^2f/DxDy = D^2f/DyDx.

From Schwarz’s theorem it follows that the Hessian matrix, H(f) is symmetric.

Question 6

What is the surface integral for cylindrical and spherical polar coordinates?

Answer 6

Cylinder:
Top: integral of F pdpd(phi) 
Base: integral of F pdpd(phi)
Curved Surface: integral of F rd(phi)dz
z is a constant for the top and base whereas for the curved surface, p=r.

Sphere: Integral of f(R, theta, phi) R^2sin(theta)d(theta)d(phi)

Question 7

What is DA when calculating the surface integral for a:
Paraboloid
Ellipsoid

Answer 7

Paraboloid: |rpxrphi|dpd(phi)
Ellipsoid: |rthetaxrphi|d(theta)d(phi)

Question 8

To find the volume of a region bounded by two surfaces, what is the surface integral we must use?

Answer 8

V= double integral of (g(x,y)-f(x,y))dA where f(x,y) is the upper bound and g(x,y) is the lower bound.

To find the region of f to integrate we must set f(x,y)=g(x,y).

Question 9

What is DV when calculating the volume integral for a:
Cylinder
Sphere

Answer 9

Cylinder: pdpd(phi)dz
Sphere: r^2sin(theta)drd(theta)d(phi)

Question 10

What is the integral for the length of a curve?

Answer 10

Integral of |r’(t)|dt where r is the position vector and t is contained in [a,b].

Question 11

What is the equation to find a line integral and line segments?

Answer 11

(1) integral of F•r’dt or the integral of Fxdxdt + Fydy/dt + Fzdz/dt) dt

(2) if x is the changing variable in the coordinates we write 
integral of (Fxdx/dx + Fydy/dx + Fzdz/dx) dx
(Fx is the x component in F whereas fx is the partial derivative of f wrt to x).

Question 12

What are the conditions for a maxima and minima ?

Answer 12

First fx=fy=0 to find the critical points.
Then do A=fxx, B=fxy, C=fyy.
AC-B^2>0 and A>0 minimum, A<0 (saddle point);
AC-B^2=0 (no information).

Question 13

How do we go about using the implicit function theorem to show we can write y=f(x)?

Answer 13

For the function F(x,y), fy must not be equal to 0, if so then we can write y=f(x).
For, x=g(y) we must show that fx is not equal to 0.
When no coordinates are given, if for example, you are trying to find y=f(x) then find fy and equate to zero and make y or x the subject then substitute this value into F which will obtain the co-ordinates at which it is not possible to write y=f(x).

Question 14

What is Green’s Theorem? And what is the Area?

Answer 14

Integral (fdx+gdy)=double integral of (dg/dx-df/dy)dxdy

Area: 0.5 x integral of (xdy-ydx)

Question 15

What is the divergence of:
F(x,y,z)
F(p,phi,z)
F(r,theta,phi)

Answer 15

F(x,y,z): divF=DFx/Dx + DFx/Dy + DFx/Dz
F(p,phi,z): divF=pD(pFp)/Dp + 1/p D(Fphi)/D(phi) + DFz/Dz
F(r,theta,phi): divF=1/r^2D(r^2Fr)/Dr + 1/rsin(theta)D(Fthetasin(theta))/D(theta) + 1/rsin(theta) D(Fphi)/Dphi

Question 16

What is Gauss’ Theorem?

Answer 16

Gauss’ Theorem States:

The (volume) integral of the divF dV = the (surface) integral of F•n dA

Question 17

What is the curl of F? (Do for cylindrical and spherical coordinates and check on sheet 6 solutions).

Answer 17

The curl is the cross product of i(D/Dx, Fx), j(D/Dy,Fy), k(D/Dz,Fz).

Question 18

What is Stoke’s Theorem?

Answer 18

The (surface) integral of curl F•n dA = the (line) integral of F•dr