Gradient, Divergence and Curl
We will cover following topics
Vector Differential Operator
The symbolic vector →i∂∂x+→j∂∂y+→k∂∂z is called Hamiltonian operator or vector differential operator and is denoted by ∇ (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
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The vector equation of the tangent plane at the point A is \left(\vec{r}-\vec{r}_{0}\right) \cdot \nabla \phi=0.
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The cartesian equation of the plane at the point A\left(x_{0}, y_{0}, z_{0}\right) is
where the partial derivatives are evaluated at the point \left(x_{0}, y_{0}, z_{0}\right).
Equation of Normal Plane
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The vector equation of the normal at the point A is \left(\vec{r}-\vec{r}_{0}\right) \times \nabla \phi=0.
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The cartesian equation of the normal at the point A is
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.
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]