Differentiation of vector fields

Differentiation of vector fields

This article is about revision notes on differentiation of vector fields. This article is not about detailed notes but these terms are important in studying physics.

1.Differentiation of vectors

Let \vec f = \hat i{f_x} + \hat j{f_y} + \hat k{f_z} be a vector where \hat i, \hat j and \hat k are its fixed unit vectors and {f_x} , {f_y} and {f_z} are the functions of time t. Now derivative of \vec f w.r.t. time t would be

(1)   \begin{equation*}  {{d\vec f} \over {dt}} = \hat i{{d{f_x}} \over {dt}} + \hat j{{d{f_y}} \over {dt}} + \hat k{{d{f_z}} \over {dt}} \end{equation*}

So derivative of vector \vec f  is also a vector whose components are derivatives of components of vector \vec f.

If vector  \vec r = \hat ix + \hat jy + \hat kz  represents the vector displacement of a moving particle from its origin at any time t then velocity vector of the particle would be the derivative of displacement vector w.r.t. time t i.e.,

(2)   \begin{equation*} \vec v = {{d\vec r} \over {dt}} = \hat i{{dx} \over {dt}} + \hat j{{dy} \over {dt}} + \hat k{{dz} \over {dt}} \end{equation*}

Similarly acceleration vector would be the derivative of velocity vector i.e.,

(3)   \begin{equation*} \vec a = {{d\vec v} \over {dt}} = \hat i{{{d^2}x} \over {d{t^2}}} + \hat j{{{d^2}y} \over {d{t^2}}} + \hat k{{{d^2}z} \over {d{t^2}}} \end{equation*}

The product of a scalar and a vector and the dot and cross product of vectors are differentiated by the ordinary calculus rules for differentiating a product provided the order of factors is kept in cross product. Thus

(4)   \begin{equation*} {{d\left( {a\vec f} \right)} \over {dt}} = {{da} \over {dt}}\vec f + a{{d\vec f} \over {dt}} \end{equation*}

(5)   \begin{equation*} {d \over {dt}}\left( {\vec f.\vec g} \right) = \vec f.{{d\vec g} \over {dt}} + {{d\vec f} \over {dt}}.\vec g \end{equation*}

(6)   \begin{equation*} {d \over {dt}}\left( {\vec f \times \vec g} \right) = \vec f \times {{d\vec g} \over {dt}} + {{d\vec f} \over {dt}} \times g \end{equation*}

For scalar triple product we have

(7)   \begin{equation*} {d \over {dt}}\left[ {\vec f.\left( {\vec g \times \vec h} \right)} \right] = {{d\vec f} \over {dt}}.\left( {\vec g \times \vec h} \right) + \vec f.\left( {{{d\vec g} \over {dt}} \times \vec h} \right) + \vec f.\left( {\vec g \times {{d\vec h} \over {dt}}} \right) \end{equation*}

And for vector triple product we have

(8)   \begin{equation*} {d \over {dt}}\left[ {\vec f \times \left( {\vec g \times \vec h} \right)} \right] = {{d\vec f} \over {dt}} \times \left( {\vec g \times \vec h} \right) + \vec f \times \left( {{{d\vec g} \over {dt}} \times \vec h} \right) + \vec f \times \left( {\vec g \times {{d\vec h} \over {dt}}} \right) \end{equation*}

2. Scalar and vector fields

We know that many physical quantities like temperature, electric or gravitational field etc. have different values at different points in space for example electric field of a point charge is large near the charge and it decreases as we go farther away from the charge. So we can say that electric field here is the physical quantity that varies from point to in space and it can be expressed as a continuous function of position of point  in that region of space .

Now term field is defined as the region in which at every point some physical quantity (temperature, electric field etc.) has a value. So by field we mean both the region and the value of physical quantity in that region.

Scalar Fields

If we consider temperature within a solid then we have a scalar field since temperature is a scalar quantity and by scalar field we mean that there are a set of values of a scalar that must be assigned throughout a continuous region of space. Again this field may be time dependent if heat is being supplied to the solid. Physical quantities like electric potential, gravitational potential, temperature, density etc. are expressed in the form of scalar fields.

Graphically scalar fields can be represented by contours which are imaginary surfaces drawn through all points for which field has same value. For temperature field the contours are called isothermal surfaces or isotherms. In case of electrostatics, if we put a point charge at any place, then electric potential around it will depend on the position of the point. Since electric potential is a scalar quantity, the field around the charge will be known as scalar potential field. If all such points at which potential is constant is joined by a surface then such a surface is called equi-potential surface.

Such surfaces are also called level surfaces, each level surface having its own constant value. Two level surfaces cannot cut each other because if they do so then scalar values corresponding to both must hold along their common line which contradicts our definition. Thus scalar point functions i.e. \varphi \left( {x,y,z} \right) are single valued functions.

Vector Fields

Like scalar fields we also have vector fields in which a vector is given for each point in space. As an example consider a fluid flowing along a tube of varying cross-section. In this case if we specify the fluid velocity at each point, we obtain a vector field, which may be dependent on time if pressure difference across the tube is varied with time.

The intensity of electric field, magnetic field and gravitational field etc. are the examples of a vector field. A vector field is represented at every point by a continuous vector function say \vec A\left( {x,y,z} \right). At any specific point of the field, the function \vec A\left( {x,y,z} \right) gives a vector of definite magnitude and direction, both of which changes continuously from point to point throughout the field region.

Graphically vector fields are represented by lines known as field or flux lines. These lines are drawn in the field in such a way that tangent at any point of the line gives the direction of vector field at that point. To express the magnitude of vector field at any point first draw an infinitesimal area perpendicular to the field line. The number of field lines passing through this area element gives the magnitude of vector field. One more important thing here to note is that, the lines representing vector fields cannot cross, because if they cross they would give non unique field direction at the point of interaction.

3. Gradient of a scalar field

Let us consider a metal bar whose temperature varies from point to point in some complicated manner. So, temperature will be a function of x, y, z in Cartesian co-ordinate system. Hence temperature here is a scalar field represented by the function T(x,y,z). Since temperature depends on distance it could increase in some directions and decrease in some directions. It could increase or decrease rapidly along some directions in comparison to other directions. Here we are interested in rate of change of temperature T with distance as we move away from starting point.

Suppose we have two points P and Q separated by a small interval \Delta \vec r as shown below in the figure.

vector differentiation

Let T1 be the temperature of point P and T2 be the temperature of point Q. So difference in temperature of point P and Q is \Delta T = {T_2} - {T_1} . Since \Delta T is scalar and does not depend on choice of co-ordinate system so we can conveniently choose a set of axis so that we could write

{T_1} = T\left( {x,y,z} \right) and {T_2} = T\left( {x + \Delta x, y + \Delta y, z + \Delta z} \right)

Where, {\Delta x}, {\Delta y} and {\Delta z} are components of vector \Delta \vec r.

By theorem of partial derivatives, the difference in values of T at two neighboring points can be written as

(9)   \begin{equation*} \Delta T = {{\partial T} \over {\partial x}}\Delta x + {{\partial T} \over {\partial y}}\Delta y + {{\partial T} \over {\partial z}}\Delta z \end{equation*}

Above equation 9 tells us how temperature T of metal bar changes when all three co-ordinates changes by infinitesimal amount {\Delta x}, {\Delta y} and {\Delta z}.

The left side of equation 9 is clearly a scalar and right side of equation 9 seems to be the dot product of two vectors and hence equation 9 can also be rewritten as

    \[\Delta T = \left( {\hat i{{\partial T} \over {\partial x}} + \hat j{{\partial T} \over {\partial y}} + \hat k{{\partial T} \over {\partial z}}} \right)  \bullet  \left( {\hat i\Delta x + \hat j\Delta y + \hat k\Delta z} \right)\]

(10)   \begin{equation*} \Delta T = \left( {\nabla T} \right) \bullet \left( {\Delta \vec r} \right) \end{equation*}

Where,

(11)   \begin{equation*} \nabla T = \left( {\hat i{{\partial T} \over {\partial x}} + \hat j{{\partial T} \over {\partial y}} + \hat k{{\partial T} \over {\partial z}}} \right) \end{equation*}

is known as gradient of T. Clearly \nabla T is a vector quantity derived from scalar field. So, equation (10) tells us that difference in temperature between two neighbouring points is the dot product of the gradient of T and the vector displacement between points.

Since gradient of any scalar field is a vector it like other vectors have magnitude as well as direction. Rewriting the dot product in equation 10 we have,

(12)   \begin{equation*} \Delta T = \nabla T \bullet \Delta \vec r = \left| {\nabla T} \right|\left| {\Delta \vec r} \right|cos\theta \end{equation*}

Where \theta is the angle between \nabla T and \Delta \vec r. If we fix the length of\Delta \vec r and let it have arbitrary direction the value of \left( {\nabla T} \right) \bullet \left( {\Delta \vec r} \right) is maximum when \Delta \vec r is along the direction of \nabla T  and it occurs when \theta  = 0 giving cos\theta = 1.

So if we move along the same direction as the gradient then function T changes maximally. So direction of \nabla T is along greatest increase of T. The magnitude |\nabla T | gives the slope along this direction as can be seen that each component of \nabla T clearly gives a slope.

4. The operator

Consider the expression 11 from the previous section which is

\nabla T = \left( {\hat i\frac{{\partial T}}{{\partial x}} + \hat j\frac{{\partial T}}{{\partial y}} + \hat k\frac{{\partial T}}{{\partial z}}} \right) = \left( {\hat i\frac{\partial }{{\partial x}} + \hat j\frac{\partial }{{\partial y}} + \hat k\frac{\partial }{{\partial z}}} \right)T

The term in the parenthesis is known as del operator ad is given by

\nabla  = \left( {\hat i\frac{\partial }{{\partial x}} + \hat j\frac{\partial }{{\partial y}} + \hat k\frac{\partial }{{\partial z}}} \right)

So, this is  is a vector operator and as such it does not have a meaning until it is made to operate on a function. We have already seen what happens when this operator acts on a scalar function say \Phi \left( {x,y,z} \right) then

\nabla \Phi \left( {x,y,z} \right) = grad \Phi

and it gives gradient of this scalar function .

This  operator acts on a vector function in two possible ways

  1. Via dot product : \nabla \cdot \vec v which gives divergence of the vector function
  2. Via cross product: \nabla \times \vec v which gives curl of a vector function.

5. Divergence and curl of a vector

Scalar product of operator \left( \nabla  \right) and a vector function {\vec v} i.e. \nabla  \cdot \vec vis known as divergence of vector . In Cartesian co-ordinates divergence of vector  can be written as

(13)   \begin{equation*} $\nabla  \cdot \vec v = \left( {\hat i\frac{\partial }{{\partial x}} + \hat j\frac{\partial }{{\partial y}} + \hat k\frac{\partial }{{\partial z}}} \right) \cdot \left( {\hat i{v_x} + \hat j{v_y} + \hat k{v_z}} \right) = \left( {\frac{{\partial {v_x}}}{{\partial x}} + \frac{{\partial {v_y}}}{{\partial y}} + \frac{{\partial {v_z}}}{{\partial z}}} \right) \end{equation*}

From equation 13 we see that divergence of a vector function is a scalar function. A vector function whose divergence vanishes is known as solenoidal vector function .

Geometrically divergence of a vector function is a measure of how much the given vector in question spreads out or diverges from point in question. Divergence of a vector function can be positive, negative or zero. If a vector spreads out from a point then it has positive divergence at that point and the point acts as source of the field {\vec A}. If the field converges at a point then divergence of vector {\vec A} will be negative at that point and the point acts as sink for field {\vec A}.

Now if we take the vector product of the operator \nabla and the vector {\vec A} then what we get is the curl of vector {\vec A}  that is

(14)   \begin{equation*} \nabla  \times \vec A = \left| {\begin{array}{*{20}{c}} {\hat i}&{\hat j}&{\hat k} \\ {\frac{\partial }{{\partial x}}}&{\frac{\partial }{{\partial y}}}&{\frac{\partial }{{\partial z}}} \\ {{A_x}}&{{A_y}}&{{A_z}} \end{array}} \right| = \hat i\left( {\frac{{\partial {A_z}}}{{\partial y}} - \frac{{\partial {A_y}}}{{\partial z}}} \right) + \hat j\left( {\frac{{\partial {A_x}}}{{\partial z}} - \frac{{\partial {A_z}}}{{\partial x}}} \right) + \hat k\left( {\frac{{\partial {A_y}}}{{\partial x}} - \frac{{\partial {A_x}}}{{\partial y}}} \right) \end{equation*}

From equation 14 we see that curl of a vector function is a vector function.

Geometrically curl of a vector function is a measure of how much the vector {\vec A}   curls around the point in question. If the curl of the vector vanishes then it is said to be irrotational. The direction of curl of vector {\vec A}   is same as the thumb of right hand when remaining fingers follow the lines of {\vec A}  .

6. Product Rules

Let φ and ψ be two scalar field functions and {\vec A} and {\vec B} be two vector fields. There are two ways to construct a scalar as a product of two functions

(a) \varphi \psi :- product of two scalar functions

(b) \vec A \cdot \vec B :- dot product of two vector functions

and there are two ways to make a vector function

(a) \phi \vec A :- scalar times a vector

(b) \vec A \times \vec B :- cross product of two vectors

Now the operator \nabla can act on a scalar function to give its gradient and on a vector function to give either its divergence or curl. Accordingly we can construct six product rules, two for gradient, two for divergence and two for curl.

a. Product rules for gradient

(i) \nabla \left( {\phi \psi } \right) = \phi \left( {\nabla \psi } \right) + \psi \left( {\nabla \phi } \right)

(ii) grad\left( {\vec A \cdot \vec B} \right) = \left( {\vec B \cdot \nabla } \right)\vec A + \left( {\vec A \cdot \nabla } \right)\vec B + \vec B \times curl\vec A + \vec A \times curl\vec B

b. Product rules for divergence

(i) \nabla  \cdot \left( {\phi \vec A} \right) = \phi \left( {\nabla  \cdot \vec A} \right) + \vec A \cdot \left( {\nabla \phi } \right)

(ii) \nabla  \cdot \left( {\vec A \times \vec B} \right) = \vec B \cdot \left( {\nabla  \times \vec A} \right) - \vec A \cdot \left( {\nabla  \times \vec B} \right)

c. Product rule for curls

(i) \nabla  \times \left( {\phi \vec A} \right) = \phi \left( {\nabla  \times \vec A} \right) - \vec A \times \left( {\nabla \phi } \right)

(ii) \nabla  \times \left( {\vec A \times \vec B} \right) = \left( {\vec B \cdot \nabla } \right)\vec A - \left( {\vec A \cdot \nabla } \right)\vec B + \vec A\left( {\nabla  \cdot \vec B} \right) - \vec B\left( {\nabla  \cdot \vec A} \right)

7. Second Derivative

Gradient of a scalar function and divergence and curl of vector functions are only first derivatives that we get by applying del operator. To obtain second derivatives we can apply the del operator twice in the following manner

a. Divergence of gradient [\nabla  \cdot \left( {\nabla \phi } \right)]

\nabla  \cdot \left( {\nabla \phi } \right) = \left( {\hat i\frac{\partial }{{\partial x}} + \hat j\frac{\partial }{{\partial y}} + \hat k\frac{\partial }{{\partial z}}} \right) \ot \left( {\hat i\frac{{\partial \phi }}{{\partial x}} + \hat j\frac{{\partial \phi }}{{\partial y}} + \hat k\frac{{\partial \phi }}{{\partial z}}} \right)

\nabla  \cdot \left( {\nabla \phi } \right) = \frac{{{\partial ^2}\phi }}{{\partial {x^2}}} + \frac{{{\partial ^2}\phi }}{{\partial {y^2}}} + \frac{{{\partial ^2}\phi }}{{\partial {z^2}}} = {\nabla ^2}\phi

Now the differential operator

div grad  = \nabla  \cdot \nabla  = {\nabla ^2}

is given a special name and is known as Laplacian operator and in Cartesian co-ordinates it is given by

{\nabla ^2} = \frac{{{\partial ^2}}}{{\partial {x^2}}} + \frac{{{\partial ^2}}}{{\partial {y^2}}} + \frac{{{\partial ^2}}}{{\partial {z^2}}}

This Laplacian operator can also cat on a vector function such that

{\nabla ^2}\vec a = \left( {{\nabla ^2}{a_x}} \right)\hat i + \left( {{\nabla ^2}{a_y}} \right)\hat j + \left( {{\nabla ^2}{a_z}} \right)\hat k

b. Curl of gradient

Curl of gradient of a scalar function is always zero i.e. \nabla  \times \left( {\nabla \phi } \right) = 0 and can easily be proved. This is true for all scalar functions.

c. Gradient of divergence [\nabla \left( {\nabla  \cdot \vec a} \right)]

Gradient of divergence of vector is not of much importance as it seldom occurs in physical applications and one important thing to note here is that it is not same as Laplacian operator.

d. Divergence of curl

Divergence of curl of a vector function is also equal to zero

\nabla  \cdot \left( {\nabla  \times \vec a} \right) = 0

e. Curl of curl

It is given by \nabla  \times \left( {\nabla  \times \vec a} \right) = \nabla \left( {\nabla  \cdot \vec a} \right) - {\nabla ^2}\left( {\vec a} \right)