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Lecture 2

5/01/2012

Prof. C.V. TOMY, email : tomy@phy.iitb.ac.in

Announcements

Tutorials start on Wednesday (11/Jan)

Webpage : www.phy.iitb.ac.in/ph103

tut sheet 1 is available on webpage

Pl take a print out before going to the tut class

mailto:tomy@phy.iitb.ac.inhttp://www.phy.iitb.ac.in/ph103http://www.phy.iitb.ac.in/ph103mailto:tomy@phy.iitb.ac.in

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Vector calculus

For a scalar function = (x),

slope

Vector operator DEL

GRADIENT of scalar function which is a vector

Gradient (vector)

For = (x,y,z), length element

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where is the angle between and

For a given dl, dT is maximum when = 0, ie, along thedirection ofTGRADIENT of a function points in the direction of max.

increase of the function

| | gives the slope along the maximum direction

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Spherical coordinates

(r, , )

cylindrical coordinates(, , z)

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Divergence (scalar)acts on a vector function through dot product

Physical meaning:

measures the spread out (divergence) of a vector at a given point

net amount of flux through a given volume

div = (incoming flux outgoing flux) from a given volume

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Spherical

coordinates

(r, , )

cylindricalcoordinates

(, , z)

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Curl (vector)

acts on a vector function through cross product

Physical meaning:

measures the circulation (curl) of a

vector at a given point

x

y

z

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Gradient TheoremFor a scalar function(x,y,z) in the interval (ra,rb)

Depends only on the end points

Independent of path

over a closed path

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Divergence theorem - contradictionsLet be a vector field . Then divergence theorem implies that

If the volume is taken as a sphere of radius R, then RHS

Hence LHS = 0

This contradiction is due to the fact that r= 0 is a singularity and

such cases should be dealt with Dirac Delta-function method of

integration.

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Curl theorem (Greens theorem)

area areaelement lengthlength

elementCurl of a vector function in a given surface isequivalent to

value of function along the bounding line enclosing the surface

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SOLID ANGLE

To define angle, circle of radius ris drawn

with the apex as its centre.

Then = L/r.L is the length of arc subtending the angleL/r is independent of the radius ( = L/r)

O

O

dS subtends at O a solidangle dd = dS cos/r2 for small dS

OTotal solid angle (for a sphericalsurface enclosing O)

OO