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Vector Differential Operator

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

∴ \nabla = \vec{i} \frac{\partial}{\partial x} + \vec{j} \frac{\partial}{\partial y} + \vec{k} \frac{\partial}{\partial z}

It is also known as del operator. This operator can be applied on a scalar point function $ϕ (x, y, z)$ or a vector point function $\vec{F} (x, y, z)$ 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 $ϕ (x, y, z)$ is a scalar point function continuously differentiable in a given region $R$ of space, then the gradient of $ϕ$ is defined by $\nabla ϕ = \vec{i} \frac{\partial ϕ}{\partial x} + \vec{j} \frac{\partial ϕ}{\partial y} + \vec{k} \frac{\partial ϕ}{\partial z}$ .

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

Directional Derivative

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

$∴$ The directional derivative $= \nabla ϕ \cdot \frac{\vec{a}}{| \vec{a} |}$ since $\nabla ϕ \cdot \frac{\vec{a}}{| \vec{a} |} = \frac{| \nabla ϕ | | \vec{a} |}{| \vec{a} |} \cos θ,$ where $θ$ is the angle between $\nabla ϕ$ and $\vec{a}$

Since $\nabla ϕ \cdot \frac{\vec{a}}{| \vec{a} |} = \frac{| \nabla ϕ | | \vec{a} |}{| \vec{a} |} \cos θ,$ where $θ$ is the angle between $\nabla ϕ$ and $\vec{a}$ $= | \nabla ϕ | \cos θ$

So, the directional derivative at a given point is maximum if $\cos θ$ is maximum. i.e., $\cos θ = 1 \Rightarrow θ = 0$ $∴$ the maximum directional derivative at a point is in the direction of $\nabla ϕ$ and the maximum directional derivative is $| \nabla ϕ |$

Equation of Tangent Plane

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

(x - x_{0}) \frac{\partial ϕ}{\partial x} + (y - y_{0}) \frac{\partial ϕ}{\partial y} + (z - z_{0}) \frac{\partial ϕ}{\partial z} = 0

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

Equation of Normal Plane

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

\frac{x - x_{0}}{\frac{\partial ϕ}{\partial x}} = \frac{y - y_{0}}{\frac{\partial ϕ}{\partial y}} = \frac{z - z_{0}}{\frac{\partial ϕ}{\partial z}}

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

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 $P$ .

θ

is the angle between the normals at the point

P

, then $$\cos \boldsymbol{\theta}=\dfrac{\nabla f \cdot \nabla g}{

\nabla f

\nabla g

}$$.

θ = \frac{π}{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$$.

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

Divergence

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

\nabla \cdot \vec{F} = \vec{i} \cdot \frac{\partial \vec{F}}{\partial x} + \vec{j} \cdot \frac{\partial \vec{F}}{\partial y} + \vec{k} \cdot \frac{\partial \vec{F}}{\partial z}

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

If $\vec{F} = F_{1} \vec{i} + F_{2} \vec{j} + F_{3} \vec{k},$ then $\nabla \cdot \vec{F} = \frac{\partial F_{1}}{\partial x} + \frac{\partial F_{2}}{\partial y} + \frac{\partial F_{3}}{\partial z}$

If $\vec{F}$ is a constant vector, then $\nabla \cdot \vec{F} = 0$ and conversely if $\nabla \cdot \vec{F} = 0,$ then $\vec{F}$ 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 $\vec{F} = 0$ everywhere in a region $R,$ then $\vec{F}$ is called a solenoidal vector point function and $R$ is called a solenoidal field.

Curl

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

\nabla \times \vec{F} = \vec{i} \times \frac{\partial \vec{F}}{\partial x} + \vec{j} \times \frac{\partial \vec{F}}{\partial y} + \vec{k} \times \frac{\partial \vec{F}}{\partial z}

It is abbreviated as curl $\bar{F}$ Thus, $curl \vec{F} = \nabla \times \vec{F}$ $\begin{array}{l} If \vec{F} = F_{1} \vec{i} + F_{2} \vec{j} + F_{3} \vec{k}, then \\ \begin{aligned} curl \vec{F} & = \nabla \times \vec{F} \\ = (\vec{i} \frac{\partial}{\partial x} + \vec{j} \frac{\partial}{\partial y} + \vec{k} \frac{\partial}{\partial z}) \times (F_{1} \vec{i} + F_{2} \vec{j} + F_{3} \vec{k}) \\ = \vec{i} [\frac{\partial F_{3}}{\partial y} - \frac{\partial F_{2}}{\partial z}] + \vec{j} [\frac{\partial F_{1}}{\partial z} - \frac{\partial F_{3}}{\partial x}] + \vec{k} [\frac{\partial F_{2}}{\partial x} - \frac{\partial F_{1}}{\partial y}] \end{aligned} \end{array}$

This is symbolically written as $\nabla \times \vec{F} = | \begin{matrix} c c c \vec{i} & \vec{j} & \vec{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ F_{1} & F_{2} & F_{3} \end{matrix} |$ .

Irrotational Vector

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

Conservative Vector

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

In a conservative vector field $\vec{F} = \nabla ϕ$

$∴ \nabla \times \vec{F} = \nabla \times \nabla ϕ = \vec{0} \Rightarrow \vec{F} is irrotational.$

PYQs

Gradient

1) Find the directional derivative of the function $x y^{2} + y z^{2} + z x^{2}$ 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 $\nabla f = \frac{\vec{r}}{r^{5}}$ 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:
$\begin{array}{l} u = x + y + z \\ v = x^{2} + y^{2} + z^{2} \\ and w = y z + z x + x y \end{array}$ .

[2011, 15M]

4) Show that the vector field defined by the vector function $\vec{v} = x y z (y z \hat{i} + x z \hat{j} + x y \hat{k})$ 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 $\frac{π}{3}$ with the $x - a x i s$ .

[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 $z - a x i s$
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 $\vec{r}$ , $r = | \vec{r} |$ , determine grad $(r^{- 1})$ 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 $z - a x i s$ .

[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 $y - a x i s$ 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 $\bar{V} = (x + y + a z) \hat{i} + (b x + 2 y - z) \hat{j} + (- x + c y + 2 z) \hat{k}$ 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 $\vec{F} = (2 x + y^{2}) \hat{i} + (3 y - 4 x) \hat{j}$ 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 $\vec{F} = (x^{2} + x y^{2}) \hat{i} + (y^{2} + x^{2} y) \hat{j}$ . 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 $\vec{F} = (2 x y + z^{3}) \hat{i} + x^{2} \hat{j} + 3 x z^{2} \hat{k}$ 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 $r^{n} \bar{r}$ 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} = 2 x y z^{3} \vecf i + x^{2} z^{3} \vec{j} + 3 x^{2} y z^{2} \vec{k}

is irrotational. Find also the scalar $u$ such that $F$ =grad $u$ .

[2001, 15M]

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