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Line, Surface and Volume Integrals

 Line Integral: The integration of a vector along a curve is called its  line integral. As shown in Figure 7.11, let MN is a curve drawn between two points M and N in vector field. Let dl is an element of length along the curve MN at O. Suppose A is the vector at 0, making an angle e with the direction of dl.

Then

 

A .dl = A dl cosθ

Dl  =  (A cosθ)

 

Equation (1) shows that the value of A.dl at any point of the curve is equal to the product of small element dl and the component (A cos 8) of A along the direction of dl. The value of A.dl for the complete curve MN can be obtained by integrating Equation (1). Hence

 

ʃNMA.dl = ʃNM dl (A cos θ).

 

 

Integral ʃNMA.dl is defined as the line integral of A along the curve MN.

 

Examples.

 

(i) If A represents the force acting on a particle moving along the curve from M to N, then the line integral ʃNMA.dl represents t1he total workdone by the force during the motion of the particle over its entire path from M to N.

 

(ii) If A denotes the electric field intensity at any point, then the line integral represents the potential difference between M and N.

 

Surface Integral

 

Consider a simple surfaceS in a vector field bounded by a curve as shown in Figure 7.12.

Let dS be an infinitesimal element of the surface. The surface element of area can be

 

represented by area vector dS. If ii be a unit positive vector (drawn outward the surface) in the· direction of dS, then

Ds =n dS

 

“Let A be a vector at middle of the element d S ID.  In the direction making an angle  θ  with . n .

Now the scalar product

A.ds = A. n ds =Ads cos θ

 

is called the flux of vector field A across the area element dS. The total flux of the vector

field across the entire surface S is given by

 

ʃʃS   A .DS =  ʃ ʃsA. n dS = ʃ ʃS A cos θ ds

 

This is defined as surface integral.

 

Volume Integral

 

Consider a closed surface in space enclosing the volume V. If A be a vector point function ata point in a small element of volume dV, then the integral

 

ʃʃʃ A dV

 

is called the volume integral of vector A over the surface.