Gradient and Divergence

Divergence

If \(\vec F (x, y, z)\) represents a vector field i.e. if \(\vec f\) be a continuous differential vector point function, the function \(\hat i .\frac {\partial \vec F }{\partial x}+ \hat j .\frac {\partial \vec F }{\partial y}+ \hat k.\frac {\partial \vec F }{\partial z} \) which is a scalar is called divergence of \(\vec F\) and is written as div \(\vec F\)

\begin{align*}\therefore div \vec F &= \hat i. \frac {\partial \vec F }{\partial x}+ \hat j. \frac {\partial \vec F }{\partial y}+ \hat k.\frac {\partial \vec F }{\partial z}\\ &= \left (\hat i \frac {\partial}{\partial x}+ \hat j \frac {\partial}{\partial y}+ \hat k\frac {\partial}{\partial z} \right )\vec F \\&= \nabla F \end{align*}

So that divergence of a vector point function \(\vec F\) is a scalar product of \(\nabla \)del operator with \( \vec F\).

Now in terms of component,\(\vec F = \hat i F_x + \hat j F_y + \hat k F_z \) then

\begin{align*}\text {div} \vec F &= \nabla . \vec F =\left (\hat i \frac {\partial }{\partial x}+ \hat j \frac {\partial}{\partial y}+ \hat k\frac {\partial}{\partial z} \right )(\hat i F_x + \hat j F_y + \hat k F_z) \\ \nabla . \vec F &= \frac {\partial F_x}{\partial x} + \frac {\partial F_y}{\partial y} + \frac {\partial F_z}{\partial z} \end{align*}

Interpretation of divergence of vector \(\vec V\)

Rate of flow of flux

Let \(\vec V\) be a vector point function representing the velocity of a fluid at any given instant ‘t’ at the middle point P inside a small parallelopiped, with edges \(\delta _x, \delta_y \text {and}\:\delta_z \) parallel to the coordinate axes parallel to the three coordinate axes.

Let \(\vec V = v_x \hat i + v_y \hat j + v_z\hat k\) be the velocity fluid at a point P inside the parallelepiped. Then the velocity of fluid on ABCD plane decreased by \(\left ( \frac {\partial V_x}{\partial x}\times \frac {\delta x}{2}\right )\) and the velocity of the fluid on A’B’C’D’ plane increase by \(\left ( \frac {\partial V_x}{\partial x}\times \frac {\delta x}{2}\right )\).

Then the velocity component along the x-axis at any point of the face ABCD normal to this axis

$$= v_x - \frac 12\frac {\partial V_x}{\partial x}\delta x+ \dots $$

[\(\therefore \) Using Taylor’s theorem and neglecting higher order \(S (x + h, y) = V (x, y) + h \frac {\partial V_(x,y)}{\partial x}\)]

Now, the mass of fluid passing per second through the face ABCD

\begin{align*} &= \text {component of velocity normal to face}\times \text {area of faces} \times \text {density of fluid} \\ &= \left (v_x - \frac 12\frac {\partial V_x}{\partial x}\delta x \right )\delta y\delta z.\rho \end{align*}

Similarly mass of fluid leaving per second through the opposite face A’B’C’D’ of the parallelepiped is
\begin{align*} = \left (v_x + \frac 12\frac {\partial V_x}{\partial x}\delta x \right )\delta y\delta z.\rho \end{align*}

Thus the volume of the flux passing through per second through the face ABCD and A’B’C’D’ of parallelepiped i.e. along the direction of x

\begin{align*} &= \left (v_x + \frac 12\frac {\partial V_x}{\partial x}\delta x \right )\delta y\delta z -\left (v_x - \frac 12\frac {\partial V_x}{\partial x}\delta x \right )\delta y\delta z \\ &= \frac {\partial v_x}{\partial x}\delta x\:\delta y\: \delta z \end{align*}

Similarly considering the other faces of parallelepiped, we have total volume of the flux moving out of the parallelepiped per second is

\begin{align*} &= \left (\frac {\partial v_x}{\partial x}+\frac {\partial v_y}{\partial y}+\frac {\partial v_z}{\partial z}\right )\delta x\:\delta y\: \delta z \\ &=\left (\hat i\frac {\partial }{\partial x}+\hat j\frac {\partial }{\partial y}+\hat j\frac {\partial }{\partial z}\right ) (\hat iv_x +\hat jv_y + \hat k v_z )\delta x\:\delta y\:\delta z\\ &= (\nabla . \vec {\nu}) dv\: \: \:[dV = \delta x\: \delta y\: \delta z]\end{align*}

Now, total gain in flux per unit volume per unit time is \((\nabla .\vec {\nu })\)

So, the amount of flux per unit volume is defined as the divergence of the vector is \( \vec {nu}\).

Solenoidal vector point function:

A vector point function \(\vec F\) is said to be a solenoidal in a region of its flux across any closed surface in that region be zero, which occurs only when div \(\vec F\)is equal to zero. For a vector to be solenoidal either the lines of flow of its flux should form closed curves or extend to infinity. For example: magnetic lines of force of the electric current.

The gradient

Let \(\phi (x, y, z)\) be defines and differentiable at each point (x, y, z) in a certain region of space . Then the gradient of \(\phi,\: \nabla \phi, \) or grad \(\phi \) is given by

\begin{align*} \nabla \phi &= \left (\frac {\partial }{\partial x} \hat i+ \frac {\partial}{\partial y} \hat j + \frac {\partial}{\partial z} \hat k\right )\phi \\&= \frac {\partial \phi}{\partial x} \hat i + \frac {\partial \phi }{\partial y} \hat j + \frac {\partial\phi }{\partial z} \hat k \\ \end{align*}

Let us consider \(\phi _1(x, y, z)\) and \(\phi_2 (x + dx, y + dy, z+ dz)\) defined in the certain region of space then difference between them is \(d\phi \). And from Taylor’s expansion,

\begin{align*} \phi _2(x + dx, y+ dy, z+dz) &= \phi_1 (x, y, z) + \left ( \frac {\partial \phi}{\partial x}dx + \frac {\partial \phi}{\partial y}dy + \frac {\partial \phi}{\partial x}dz\right ) + \dots \\ \phi _2(x + dx, y+ dy, z+dz) -\phi_1 (x, y, z) &= \frac {\partial \phi}{\partial x}dx + \frac {\partial \phi}{\partial y}dy + \frac {\partial \phi}{\partial x}dz \end{align*}

\begin{align*} \text {or,}\: d\phi &= \left (\frac {\partial \phi}{\partial x} \hat i + \frac {\partial \phi }{\partial y} \hat j + \frac {\partial \phi }{\partial z} \hat k \right ). (\hat i dx + \hat j dy+ \hat k dz) \\ &=\left (\frac {\partial }{\partial x} \hat i+ \frac {\partial}{\partial y} \hat j + \frac {\partial}{\partial z} \hat k\right )\phi .\: d\vec r \end{align*}

\begin{align*} \therefore d\phi =\vec {\nabla }\phi.\: d\vec r \: \text {where}\:\vec {\nabla } = \frac {\partial}{\partial x} \hat i + \frac {\partial}{\partial y} \hat j + \frac {\partial}{\partial z} \hat k \end{align*}

Bibliography

P.B. Adhikari, Bhoj Raj Gautam, Lekha Nath Adhikari. A Textbook of Physics. kathmandu: Sukunda Pustak Bhawan, 2011.

Jha, V. K.; 'Lecture title'; Elementary Vector Analysis; St. Xavier's College, Kathmandu; 2016.