1. Partial di¤erential and gradient of a scalar potential U = U (~r) = U (x; y; z) , ~r = x^i + y^ j + z^ k

Mathematics for physicists

Ven-Chung Lee

(Dated: October 28, 2006)

1. Partial di¤erential and gradient of a scalar potential U = U (~r) = U (x; y; z) , ~r = x^i + y^ j + z^ k

, ˆ dr s r ds

r r

total di¤erential

dU = U (~r + d~r) U (~r) = @U

@x dx + @U

@y dy + @U

@z dz

@U

@x = lim

x!0

U (x + x; y; z) U (x; y; z) x

dU = @U

@x ^i + @U

@y

^ j + @U

@z

^ k dx^i + dy^ j + dz^ k = ~ rU d~r rU (~r) = ^i ~ @U

@x + ^ j @U

@y + ^ k @U

@z gradient of U at point ~r, gradient …eld

dU = ~ rU d~r

ds ds = ~ rU ^sds ) dU

ds = ~ rU ^s directional derivative of U along ^ s direction 2. Gradient and equi-potential surface

n ˆ

( )

U r r = U t ˆ

rr

U = U (~r) = U

= const:

equi-potential surface, equi-pressure, equi-temperature

2

if d~r k ^t, ^t :any tangent vector of the surface on the point ~r, then

dU = 0 = ~ rU ^t

) ~rU k ^n , ^n : normal vector of the surface ) ^n = rU (~r) ~

rU (~r) ~

This relation provides a way to …nd the normal vector. If d~r = ^ nd` then rU (~r) = ~ dU

d` ^ n .

3. Gradient in the spherical coordinates if ~r = (r; ; ') then

d~r = dr ^ r + rd ^ + r sin d' ^ ' ) dU = @U

@r dr + @U

@ d + @U

@' d' = @U

@r ^ r + 1 r

@U

@ ^ + 1 r sin

@U

@' ' ^ d~r ) ~rU (r; ; ') = @U

@r ^ r + 1 r

@U

@ ^ + 1 r sin

@U

@' ' ^ 4. Line integral of a vector …eld

O 1

2

rr F r

drr L

W

= Z

1 L

F d~r ~

Work done by force ~ F from position 1 to 2 along the line L Tangent vector

^ t = d~r

ds , ds = jd~rj along L W =

Z

F ^ ~ tds = Z

F

ds On x-y plane

Z

F d~r = ~ Z

(F

dx + F

dy) , F = F ~

^{ + F

^ |

3

5. Conservative …eld If the line integral is indep. of the path: R

F d~r = ~ R

F d~r, then ~

0

@ Z

+ Z

1 L2

1

A ~ F d~r = 0 1

2 L

L

or I

L1+L2

F d~r = 0 ~ circulation of ~ F around a loop

! ~ F (~r) is conservative or irrotational ! gravitational …eld electrostatic …eld 6. Conservation of mechanical energy If R

1

F d~r ~ is indep. of the path, and ~ F = rU ~ then

Z

F d~r = ~ Z

rU d~r = ~ Z

1

dU (~r) = [U (~r

) U (~r

)] . The line integral R

1

F d~r ~ is indep. of the path, and dep. on U (~r

) and U (~r

) only.

) a conservative …eld is a gradient …eld.

W

= Z

1

F d~r = U (~r ~

) U (~r

)

If ~ F is the force then U is the potential energy,

* Z

F d~r = ~ Z

m d~v dt ~r =

Z

md~v ~v = 1 2 m~v ~v ) Z

1

F d~r = W ~

= 1

2 mv

1

2 mv

= U (~r

) U (~r

) ) 1

2 mv

+ U (~r

) = 1

2 mv

+ U (~r

) Conservation of mechanical energy 7. Eq. of di¤usion, heat conduction, electric conduction

U ! density, temperature, electric potential electric current density by drift

J = q ~ E ~ Ohm’s law

: mobility , E = ~ rV : electric …eld , V : electric potential ~

4 electric current density by di¤usion

J = ~ q D ~ r

D : di¤usion constant , q : charge of carrier , : carrier density heat current

J = ~ rT , ~ : heat conductivity one-dimensional case

J = J ^{ = ~ E = ~ dV dx ! I

A = V

` , I : current

= ) I = V

R , R = `

A : resistance

l

1 2

A

8. Green’s theorem If ~ F = ~ F (x; y)

y + dy y

x x dx + l

x y

S Loop L

I

Loop

F d~r = ~ I

(F

dx + F

dy)

= X

i

I

`i

F d~r ~

the sum of the line integrals of all the di¤erential loops I

`i

F d~r = F ~

(x; y) dx F

(x; y + dy) dx + F

(x + dx; y) dy F

(x; y) dy

= @F

@y + @F

@y dxdy I

L

F d~r = ~ I

L

(F

dx + F

dy) = ZZ

S

@F

@x

@F

@y dxdy

| {z }

area element

= ) Green’s theorem S is the area bounded by the loop L.

9. If I

F d~r = 0 ~ for any loop on x-y plane, then

=

for any point ~r. This is the

condition for a conservative …eld ~ F on x-y plane. Further examples will be discussed

in the coming course.