1.    Line Integral. Consider a vector field defined by F = F (x, y, z). Let AB be a smooth curve drawn in this vector field. At any point P of this curve, the value of vector is F. Consider a small element of length dr at P. At P the vector F is inclined at any angle θ to dr. If F varies in magnitude and direction from point to point along the curve, then the integral.
                 ∫ F. dr = ∫ F cos θ dr
is defined as the line integral of F along the curve AB.

2.    Surface Integral. Let S be a closed surface drawn in a vector field. dS in an infinitesimal element of the surface. Let n be a unit positive vector normal to dS. The surface elements of area dS is represented by a vector dS whose magnitude is dS and whose direction is that of n. Thus dS = n dS
dS is called the area vector.

Now if A be a vector at the middle of the element dS in a director making an angle θ with n, then the integral.
        ∫∫ A . dS = ∫∫ A .n dS = ∫∫ A cos θ dS
is defined as the ‘surface integral’ or ‘flux’ of A across the surface S.



Examples. Let A denote the velocity of a moving fluid in which is fixed surface S is drawn. Then the surface integral ∫∫ A. dS gives the amount of fluid flowing per unit time normally through the surface S.

3.    Volume integral. Consider a closed surface in spaces enclosing a volume V. Let A be a vector point function at a point in a small element dV of the region. Then the integral ∫∫ ∫AdV is called the ‘volume integral’ of A ove the volume V.


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