Calculus, Theorems

7/30/2019 Calculus, Theorems

1/18

Second Derivatives The gradient, the divergence and the curl are the

only first derivatives we can make with , by

applying twice we can construct five speciesof second derivatives.

The gradient is a vector, so we can take thedivergence and curl of it.

(1) Divergence of gradient : (Laplacian)(2) Curl of gradient:

o The divergence is a scalar, so we can take itsgradient.

(3) Gradient of divergence.o The curl is a vector, so we can take its divergence

and curl.

(4) Divergence of a Curl.

(5) Curl of curl.

0)( A

).( A

0).( A

AAA

2

).()(

AA2

).(


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Integral Calculus

oLine ( Path) Integrals

o Surface (Flux) Integrals

o Volume Integrals


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Line ( Path) Integrals:

Let v is a vector function

dl is the infinitesimal displacement vectorIntegral carried out along a specified path from

point a to point b gives line or path integral.

Line integral depends critically on the particular

path taken from a to b.

.

b

av dl


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There is a special type of vector which does notdepend on path, but is determined entirely by the end

points.

The force which have this property called as

conservative force.

If path forms a closed loop i.e. a = b, the integral

can be written as:

.v dl


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Surface (Flux) Integrals:

Let v is a vector functionda is an infinitesimal patch of area (direction is

to the surface)

If v is flow of a fluid (mass per unit area per unit

time)

.Sv da: total mass per unit

time

passing through

the surface


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If the surface is closed:

oFor closed surface, direction of da is outwardo For open surfaces its arbitrary.

oSurface integral depends on the particular surfacechosen.

oThere is a special class of vector function for

which integral is independent of the surface, and is

determined entirely by theboundary line.

.v da


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Volume Integrals:

Let T is a scalar function. (may be a vector also)d is an infinitesimal volume element

,in Cartesian coordinates: d =dx.dy.dz

If T is the density of a substance

: totalmass

Note: If T is a vector function then unit vectors can

be taken out from the integral

v

Td

v Td


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The fundamental theorem of calculus:

( ) ( ) ( )

b b

a a

dfdx F x dx f b f adx

Where df/dx=F(x), we can think f(x) is a function whose

derivative is F(x)

and df=(df/dx).dx is the infinitesimal change in f when wego from x to x+dx

Integral of a derivative over an interval (a to b) is given by

the value of the function at the end points (boundaries)


9/18

In vector calculus: three species of derivatives

oGradient

oDivergent

oCurl

Each has its own fundamental

theorem


10/18

The fundamental theorem of Gradient

o

Suppose we have a scalarfunction of three variables:

T(x, y, z)

o By moving a distance dl1,

the function T will change

by an amount:1

( ).dT T dl

o So total change in T in going from a to

b along the path selected: ( ).

b

a

T dl


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o Also total change in T by going from a

to b can be represented by : T b T a

Or we can write:

This is called fundamental theorem of Gradient.

It says The line integral of a derivative (gradient) is

given by the value of the function at the boundaries.

( ). ( ) ( )

b

a

T dl T b T a


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( ). ( ) ( )

b

a

T dl T b T a

Left hand side tells: integral is path dependent

Right hand side tells: integral is path independent

(depend only on end points)

Generally line integrals depend on path taken.

Note:But Gradients have the special property that

their integrals are path independent.


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Ques: If we are choosing a closed path, then change inthe scalar function using Gradient theorem will be:

( ).T dl 0


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The Fundamental Theorem for Divergences

known as Gauss Divergence Theorem

o It general interpretation (not complete): Theintegral of a derivative over a region is equal to

the value of the function at the boundary.

( . ) .v Sv d v da


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oComplete description:

Derivative is the form of (divergence)

Region is (volume)

Boundary is the surface that bounds the volume

(boundary indicates integral not just the

difference of two points).

Note:

The boundary of a line:

The boundary of a volume:

2 end point

Closed surface


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Its Geometrical Interpretation:If v: flow of a incompressible fluid

The flux of v: = total amount of fluid

passing out through the surface, per unit time

Applications:

GausssTheorem can be applied to any vector fieldwhich obeys an inverse-square law, such as

gravitational, electrostatic attraction, and evenexamples in quantum physics such as probability

density.

( . )v

v d


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The fundamental theorem for Curls

known as Stokes Theorem

( ). .S P

v da v dl

o It general interpretation (not complete): The

integral of a derivative over a region is equal to the

value of the function at the boundary.

Line integral

o Note: on the right hand side circle over integral

indicates perimeter of the surface is closed, not

the surface itself


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oComplete description:

Derivative is the form of (curl)

Region is (patch of surface)

Boundary is the perimeter of the patch (due to perimeter

hereboundary indicates a closed line integral).

Outcomes:

1) depends only on the boundary line, not on

the particular surface used2) =0 for any closed surface, sincefor a closed

surface, boundary line shrinks down to a point.

( ).S v da( ).v da

Calculus, Theorems

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