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Vector Differential Operator

The symbolic vector $\vec{i} \dfrac{\partial}{\partial x}+\vec{j} \dfrac{\partial}{\partial y}+\vec{k} \dfrac{\partial}{\partial z}$ is called Hamiltonian operator or vector differential operator and is denoted by $\nabla$ (read as del or nabla).

\[\therefore\boldsymbol{\nabla}=\vec{i} \dfrac{\partial}{\partial \boldsymbol{x}}+\vec{j} \dfrac{\partial}{\partial y}+\vec{k} \dfrac{\partial}{\partial z}\]

It is also known as del operator. This operator can be applied on a scalar point function $\phi(x, y, z)$ or a vector point function $\overrightarrow{\mathrm{F}}(x, y, z)$ which are differentiable functions. This gives rise to three field quantities namely gradient of a scalar, divergence of a vector and curl of a vector function.

Gradient

If $\phi(x, y, z)$ is a scalar point function continuously differentiable in a given region $R$ of space, then the gradient of $\phi$ is defined by $\nabla \boldsymbol{\phi}=\vec{i} \dfrac{\partial \boldsymbol{\phi}}{\partial x}+\vec{j} \dfrac{\partial \boldsymbol{\phi}}{\partial y}+\vec{k} \dfrac{\partial \boldsymbol{\phi}}{\partial z}$.

$\nabla \phi$ is normal to the surface $\phi(x, y, z)=C$ at $P$. A unit normal to the surface at $P$ is $\vec{n}=\dfrac{\nabla \phi}{\vert \nabla \phi \vert}$.

Directional Derivative

The directional derivative of a scalar point function $\phi$ in a given direction $\vec{a}$ is the rate of change of $\phi$ in that direction. It is given by the component of $\nabla \phi$ in the direction of $\vec{a}$.

$\therefore$ The directional derivative $=\nabla \boldsymbol{\phi} \cdot \dfrac{\vec{a}}{\vert \vec{a} \vert}$ since $\nabla \phi \cdot \dfrac{\vec{a}}{\vert \vec{a} \vert}=\dfrac{\vert \nabla \phi \vert\vert \vec{a} \vert}{\vert \vec{a} \vert} \cos \theta,$ where $\theta$ is the angle between $\nabla \phi$ and $\vec{a}$

Since $\nabla \boldsymbol{\phi} \cdot \dfrac{\vec{a}}{\vert \vec{a} \vert}=\dfrac{\vert \nabla \phi \vert\vert \vec{a} \vert}{\vert \vec{a} \vert} \cos \theta,$ where $\theta$ is the angle between $\nabla \phi$ and $\vec{a}$ $=\vert \nabla \phi \vert \cos \theta$

So, the directional derivative at a given point is maximum if $\cos \theta$ is maximum. i.e., $\cos \theta=1 \Rightarrow \quad \theta=0$ $\therefore$ the maximum directional derivative at a point is in the direction of $\nabla \phi$ and the maximum directional derivative is $\vert \nabla \phi \vert$

Equation of Tangent Plane

The vector equation of the tangent plane at the point $A$ is $\left(\vec{r}-\vec{r}_{0}\right) \cdot \nabla \phi=0$.
The cartesian equation of the plane at the point $A\left(x_{0}, y_{0}, z_{0}\right)$ is

\[\left(x-x_{0}\right) \dfrac{\partial \phi}{\partial x}+\left(y-y_{0}\right) \dfrac{\partial \phi}{\partial y}+\left(z-z_{0}\right) \dfrac{\partial \phi}{\partial z}=0\]

where the partial derivatives are evaluated at the point $\left(x_{0}, y_{0}, z_{0}\right)$.

Equation of Normal Plane

The vector equation of the normal at the point $A$ is $\left(\vec{r}-\vec{r}_{0}\right) \times \nabla \phi=0$.
The cartesian equation of the normal at the point $A$ is

\[\dfrac{x-x_{0}}{\dfrac{\partial \phi}{\partial x}}=\dfrac{y-y_{0}}{\dfrac{\partial \phi}{\partial y}}=\dfrac{z-z_{0}}{\dfrac{\partial \phi}{\partial z}}\]

where the partial derivatives are evaluated at $\left(x_{0}, y_{0}, z_{0}\right)$.

Angle between Two Surfaces at a Common Point

The angle between two surfaces $f(x, y, z)=C_{1}$ and $g(x, y, z)=C_{2}$ at a common point $P$ is the angle between their normals at the point $P$.

If $\theta$ is the angle between the normals at the point $P$, then $$\cos \boldsymbol{\theta}=\dfrac{\nabla f \cdot \nabla g}{

\nabla f

\nabla g

}$$.

If $\boldsymbol{\theta}=\dfrac{\boldsymbol{\pi}}{2},$ then the normals are perpendicular and $$\cos \boldsymbol{\theta}=0 \Rightarrow \dfrac{\nabla f \cdot \nabla g}{

\nabla f | \nabla g

}=0 \Rightarrow \nabla f \cdot \nabla g=0$$.

$\therefore$ if two surfaces are orthogonal at the point $P$ then $\nabla f \cdot \nabla g=0$.

Divergence

If $\overrightarrow{\mathrm{F}}(x, y, z)$ be a vector point function continuously differentiable in a region $R$ of space, then the divergence of $\overrightarrow{\mathrm{F}}$ is defined by

\[\nabla \cdot \overrightarrow{\mathrm{F}}=\vec{i} \cdot \dfrac{\partial \overrightarrow{\mathrm{F}}}{\partial x}+\vec{j} \cdot \dfrac{\partial \overrightarrow{\mathrm{F}}}{\partial y}+\vec{k} \cdot \dfrac{\partial \overrightarrow{\mathrm{F}}}{\partial z}\]

It is abbreviated as $\operatorname{div} \overrightarrow{\mathrm{F}}$ and thus, $\operatorname{div} \overrightarrow{\mathrm{F}}=\nabla \cdot \overrightarrow{\mathrm{F}}$

If $\overrightarrow{\mathrm{F}}=\mathrm{F}_{1} \vec{i}+\mathrm{F}_{2} \vec{j}+\mathrm{F}_{3} \vec{k}, \quad$ then $\quad \nabla \cdot \overrightarrow{\mathrm{F}}=\dfrac{\partial \mathrm{F}_{1}}{\partial x}+\dfrac{\partial \mathrm{F}_{2}}{\partial y}+\dfrac{\partial \mathrm{F}_{3}}{\partial z}$

If $\overrightarrow{\mathrm{F}}$ is a constant vector, then $\nabla \cdot \overrightarrow{\mathrm{F}}=0$ and conversely if $\nabla \cdot \overrightarrow{\mathrm{F}}=0,$ then $\overrightarrow{\mathrm{F}}$ is a constant vector.

Physical interpretation of divergence applied to a vector field is that it gives approximately the ‘loss’ of the physical quantity at a given point per unit volume per unit time.

Solenoidal Vector

If div $\overrightarrow{\mathrm{F}}=0$ everywhere in a region $R,$ then $\overrightarrow{\mathrm{F}}$ is called a solenoidal vector point function and $R$ is called a solenoidal field.

Curl

If $\overrightarrow{\mathrm{F}}(x, y, z)$ be a vector point function continuously differentiable in a region $R,$ then the curl of $\overrightarrow{\mathrm{F}}$ is defined by

\[\nabla \times \overrightarrow{\mathbf{F}}=\vec{i} \times \dfrac{\partial \overrightarrow{\mathbf{F}}}{\partial x}+\vec{j} \times \dfrac{\partial \overrightarrow{\mathbf{F}}}{\partial y}+\vec{k} \times \dfrac{\partial \overrightarrow{\mathbf{F}}}{\partial z}\]

It is abbreviated as curl $\bar{F}$ Thus, $\operatorname{curl} \overrightarrow{\mathbf{F}}=\nabla \times \overrightarrow{\mathbf{F}}$ $\begin{array}{l} \text { If } \overrightarrow{\mathrm{F}}=\mathrm{F}_{1} \vec{i}+\mathrm{F}_{2} \vec{j}+\mathrm{F}_{3} \vec{k} \text { , then } \\ \qquad \begin{aligned} \text { curl } \overrightarrow{\mathrm{F}} &=\nabla \times \overrightarrow{\mathrm{F}} \\ &=\left(\vec{i} \dfrac{\partial}{\partial x}+\vec{j} \dfrac{\partial}{\partial y}+\vec{k} \dfrac{\partial}{\partial z}\right) \times\left(\mathrm{F}_{1} \vec{i}+\mathrm{F}_{2} \vec{j}+\mathrm{F}_{3} \vec{k}\right) \\ &=\vec{i}\left[\dfrac{\partial \mathrm{F}_{3}}{\partial y}-\dfrac{\partial \mathrm{F}_{2}}{\partial z}\right]+\vec{j}\left[\dfrac{\partial \mathrm{F}_{1}}{\partial z}-\dfrac{\partial \mathrm{F}_{3}}{\partial x}\right]+\vec{k}\left[\dfrac{\partial \mathrm{F}_{2}}{\partial x}-\dfrac{\partial \mathrm{F}_{1}}{\partial y}\right] \end{aligned} \end{array}$

This is symbolically written as $\nabla \times \overrightarrow{\mathrm{F}}=\begin{vmatrix}{ccc}\vec{i} & \vec{j} & \vec{k} \\ \dfrac{\partial}{\partial x} & \dfrac{\partial}{\partial y} & \dfrac{\partial}{\partial z} \\ \mathrm{~F}_{1} & \mathrm{~F}_{2} & \mathrm{~F}_{3}\end{vmatrix}$.

Irrotational Vector

Let $\overrightarrow{\mathrm{F}}(x, y, z)$ be a vector point function. If curl $\overrightarrow{\mathrm{F}}=\overrightarrow{0}$ at all points in a region $R,$ then $\overrightarrow{\mathrm{F}}$ is said to be an irrotational vector in $R$. The vector field $R$ is called an irrotational vector field.

Conservative Vector

A vector field $\overrightarrow{\mathrm{F}}$ is said to be conservative if there exists a scalar function $\phi$ such that $\overrightarrow{\mathrm{F}}=\nabla \boldsymbol{\phi}$.

In a conservative vector field $\overrightarrow{\mathrm{F}}=\nabla \boldsymbol{\phi}$

$\therefore \nabla \times \overrightarrow{\mathrm{F}}=\nabla \times \nabla \boldsymbol{\phi}=\overrightarrow{0} \Rightarrow \overrightarrow{\mathrm{F}} \text { is irrotational. }$

PYQs

Gradient

1) Find the directional derivative of the function $xy^2+yz^2+zx^2$ along the tangent to the curve $x=t$, $y=t^2$, $z=t^3$ at the point $(1,1,1)$.

[2019, 10M]

2) Find $f(r)$ such that $\nabla f=\dfrac{\vec{r}}{r^{5}}$ and $f(1)=0$.

[2016, 10M]

3) Examine whether the vectors $\nabla u$, $\nabla u$ and $\nabla w$ are coplanar, where $u$, $v$ and $w$ are the scalar functions defined by:
$\begin {array}{l}{u=x+y+z} \\ {v=x^{2}+y^{2}+z^{2}} \\ {\text { and } w=y z+z x+x y}\end{array}$.

[2011, 15M]

4) Show that the vector field defined by the vector function $\vec{v}=x y z(yz \hat{i}+ xz \hat{j}+ xy\hat{k})$ is conservative.

[2010, 12M]

5) Find the directional derivative of $f(x, y)=x^{2} y^{3}+x y$ at the point $(2,1)$ in the direction of a unit vector which makes an angle of $\dfrac{\pi}{3}$ with the $x-axis$.

[2010, 10M]

6) Find the directional derivative of
i) $4 x z^{3}-3 x^{2} y^{2} z^{2}$ at $(2,-1,2)$ along $z-axis$
ii) $-x^{2} y z+4 x z^{2}$ at $(1,-2,1)$ in the direction of $2 \hat{i}-\hat{j}-2 \hat{k}$.

[2009, 12M]

7) If $\vec{r}$ denotes the position vector of a point and if $\hat{r}$ be the unit vector in the direction of $\vec{r}$, $r=\vert \vec{r}\vert$, determine grad $\left(r^{-1}\right)$ in terms of $\hat{r}$ and $r$.

[2007, 12M]

8) Find the values of constants $a$, $b$ and $c$ so that the directional derivative of the function $f=a x y^{2}+b y z+c z^{2} x^{2}$ at the point $(1,2,-1)$ has maximum magnitude 64 in the direction parallel to $z-axis$.

[2006, 12M]

9) Find the values of constants $a$, $b$ and $c$ such that the maximum value of directional derivative of $f=a x y^{2}+b y z+c x^{2} z^{2}$ at $(1,-1,1)$ is in the direction parallel to $y-axis$ and has magnitude 6.

[2002, 15M]

10) Find the directional derivative of $f=x^{2} y z^{3}$ along $x=e^{-t}$, $y=1+2 \sin t$, $z=t-\cos t$ at $t=0$.

[2001, 15M]

Divergence

1) For what values of the constant $a$, $b$ and $c$ the vector $\overline{V}=(x+y+a z) \hat{i}+(b x+2 y-z) \hat{j}+(-x+c y+2 z) \hat{k}$ is irrotational. Find the divergence in cylindrical coordinates of the vector with these values.

[2017, 10M]

2) Prove that the divergence of a vector field is invariant with respect to co-ordinate transformations.

[2003, 12M]

Curl

1) Find the circulation of $\vec{F}$ round the curve $C$, where $\vec{F}=(2x+y^2)\hat{i}+(3y-4x)\hat{j}$ and $C$ is the curve $y=x^2$ from $(0,0)$ to $(1,1)$ and the curve $y^2=x$ from $(1,1)$ to $(0,0)$.

[2019, 15M]

2) A vector field is given by $\vec{F}=\left(x^{2}+x y^{2}\right) \hat{i}+\left(y^{2}+x^{2} y\right) \hat{j}$. Verify that the field is irrotational or not. Find the scalar potential.

[2015, 12M]

3) If $\vec{r}$ be the position vector of a point, find the value(s) of $n$ for which the vector $r^{n} \vec{r}$ is:
i) irrotational,
ii) solenoidal.

[2011, 15M]

4) Show that $\vec{F}=\left(2 x y+z^{3}\right) \hat{i}+x^{2} \hat{j}+3 x z^{2} \hat{k}$ is a conservative force field. Find the scalar potential for $\vec{F}$ and the work done in moving an object in this field from $(1-2,1)$ to $(3,1,4)$.

[2008, 12M]

5) For any constant vector, show that the vector $\vec{a}$ represented by curl $(\vec{a} \times \vec{r})$ is always parallel to the vector $\vec{a}$, $\vec{r}$ being the position vector of a point $(x, y, z)$ measured from the origin.

[2007, 15M]

6) If $\vec{r}=x \hat{i}+y \hat{j}+x \hat{k}$, find the value(s) of in order that $r^{n} \vec{r}$ may be:
i) solenoidal
ii) irrotational.

[2007, 15M]

7) Prove that $r^{n} \overline{r}$ is an irrotational vector for any value of $n$ but is solenoidal only if $n+3=0$.

[2006, 15M]

8) Prove that the curl of a vector field is independent of the choice of coordinates.

[2005, 12M]

9) Show that if $\vec{A}$ and $\vec{B}$ are irrotational, then $\vec{A} \times \vec{B}$ is solenoidal.

[2004, 12M]

10) Show that the vector field defined by

\[\vec{F} = 2xyz^3 \vecf{i}+ x^2z^3 \vec{j} + 3x^2yz^2 \vec{k}\]

is irrotational. Find also the scalar $u$ such that $\mathbf{F}$=grad $u$.

[2001, 15M]

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