Vector Calculus: Finding out divergence and curl of vector field

Vector Calculus – Finding out divergence and curl of vector field:

In case of fluid flow,

Divergence:  Relates to the Way in which fluid flows away
             or, toward from a point. 
             It has Scaler values.

Curl:        Rotational properties of a fluid at a point.
             It has vector values.

Vector field in 3 space with xyz co-ordinate system,
\overrightarrow{F}(x,y,z) = < f(x,y,z), g(x,y,z), h(x,y,z) >

Then divergence of function \overrightarrow{F} (x, y, z) is,
div \vec{F} = \nabla . \vec{F} = < \frac{\delta}{\delta x}, \frac{\delta}{\delta y}, \frac{\delta}{\delta z} > . < f, g, h >
div \vec{F} = \frac{\delta f}{\delta x} + \frac{\delta g}{\delta y} + \frac{\delta h}{\delta z}

The curl of the function \vec{F} (x, y, z) is,
curl \vec{F} = \nabla \times \vec{F}

= \begin{vmatrix}  \hat{i} & \hat{j} & \hat{k} \\  \frac{\delta}{\delta x} & \frac{\delta}{\delta y} & \frac{\delta}{\delta z} \\  f & g & h  \end{vmatrix}

curl \vec{F} = < \frac{\delta h}{\delta y} - \frac{\delta g}{\delta z}, \frac{\delta f}{\delta z} - \frac{\delta h}{\delta x}, \frac{\delta g}{\delta x} - \frac{\delta f}{\delta y} >


Example:

curl \vec{F} = < x^2y, 2y^3z, 3z >

Here,
f(x,y,z) = x^2y \\  f(x,y,z) = 2y^3z \\  f(x,y,z) = 3z

We know divergence function,
div \vec{F} = \frac{\delta f}{\delta x} + \frac{\delta g}{\delta y} + \frac{\delta h}{\delta z} \ \ \ \ \ (1)

Now,
\frac{\delta f}{\delta x} = \frac{\delta}{\delta x} x^2y = 2xy \\  \\  \frac{\delta g}{\delta y} = \frac{\delta}{\delta y} 2y^3z = 6y^2z \\  \\  \frac{\delta h}{\delta z} = \frac{\delta}{\delta z} 3z = 3

So, from eqn(1) we get,
div \vec{F} = 2xy + 6y^2z + 3

Also, The curl of the function,
curl \vec{F} = \nabla \times \vec{F}
= \begin{vmatrix}  \hat{k} & \hat{k} & \hat{k} \\  \frac{\delta}{\delta x} & \frac{\delta}{\delta y} & \frac{\delta}{\delta z} \\  f & g & h  \end{vmatrix}
curl \vec{F} = < \frac{\delta h}{\delta y} - \frac{\delta g}{\delta z}, \frac{\delta f}{\delta z} - \frac{\delta h}{\delta x}, \frac{\delta g}{\delta x} - \frac{\delta f}{\delta y} > \ \ \ \ \ (2)

calculating partial derivatives,
\frac{\delta h}{\delta y} = \frac{\delta}{\delta y} 3z = 0 \\  \\  \frac{\delta g}{\delta z} = \frac{\delta}{\delta z} 2y^3z = 2y^3 \\  \\  \frac{\delta f}{\delta z} = \frac{\delta}{\delta z} x^2y = 0 \\  \\  \frac{\delta h}{\delta x} = \frac{\delta}{\delta x} 3z = 0 \\  \\  \frac{\delta g}{\delta x} = \frac{\delta}{\delta x} 2y^3z = 0 \\  \\  \frac{\delta f}{\delta y} = \frac{\delta}{\delta y} x^2y = x^2

So, from eqn(2) we write,
curl \vec{F} = < 0 - 2y^3, 0 - 0, 0 - x^2 > \\  \\  curl \vec{F} = < -2y^3, 0, - x^2 >  \\  or,  \\  curl \vec{F} = -2y^3 \hat{i} - x^2 \hat{k};

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