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Thursday, January 14, 2010

Vector Differentiation 2

In this post we'll discuss about Divergence and curl of vector fields.
Divergence of vector fields
We already know about dot product of two vectors. Now consider a vector field φ(x,y,z) and we now find out the dot product of operator ∇ and vector field φ(x,y,z). So,


Thus we see that dot product of del operator with a vector field is a scalar quantity. This scalar quantity ∇⋅φ(x,y,z) is known as divergence of the vector function φ(x,y,z). Thus,
∇⋅φ = div φ='divergence of φ'
Physically we can say that divergence of any vector field is the measure of how much the vector diverges (spreads out) or converges at a given point. Divergence of a vector at any point can be positive, negative and zero illustrated below in the figures.

Consider the example of a fluid, if the Divergence is positive at any point in a fluid , then either the fluid is expanding and its density at the point is falling with time or the point is a source of the fluid. Again if the divergence is negative , then either the fluid is contracting and the density is rising at the point or the point is at the sink. If the flux entering field space is equal to that leaving it then divergence of the field is zero i.e. there is no source or sink and its density is also not changing which means that liquid is incompressible. Any vector having its divergence equal to zero is known as solenoidal vector.
Curl of the vector fields
We have already discussed the dot product of del operator (∇) with a vector which is known as divergence of vector. Now let us find the cross product of operator ∇ with the vector field φ(x,y,z). Thus,


Thus we have seen that curl of a vector function is a vector whose components can be written by usual rule of cross products. Any vector field φ(x,y,z) for which ∇xφ(x,y,z) = 0 is said to be irrotational.The curl of a scalar field makes no sense.The curl of a vector field φ(x,y,z) at a point P may be regarded as a measure of the circulation or how much the field curls around a point.
In our next discussion we'll learn about second derivatives of vector fields and we'll discuss more about divergence and curl while discussing vector integral calculas.

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