2 Vector differentiation

TABLE OF CONTENTS

1. Differentiation with respect to a scalar

2. Curves in space

3. Kinematics

4. The gradient

4.1 Geometrical interpretation

4.2 

4.3 

4.4 The directional derivative

5. The divergence, curl and Laplacian operators

5.1 The divergence

5.2 The curl

5.3 The product rules

5.4 The Laplacian

 

 

LEARNING OBJECTIVES

  1. Simple differentiation of a vector with respect to a scalar, like time, is discussed.
  2. Next the description of a curve in space is given and the concept of curvature and radius of curvature is discussed.
  3. The kinematics of the motion of a particle in vector form is described.
  4. Derivative of a vector with respect to space coordinates, the gradient, is introduced. Geometrical interpretation of the gradient is also discussed.
  5. Properties of the gradient, both as a differential operator and as a vector are given.
  6. Related differential operators of divergence and curl of a vector and the Laplacian operator are described.
  7. The product rules when gradient, divergence and curl of more than one scalar or vector functions is involved are written down.

V e c t o r  D i f f e r e n t i a t i o n

 

1. Differentiation with respect to a scalar

 

Differentiation of a vector with respect to a scalar is akin to ordinary differentiation of a function of one variable.

        

In terms of the Cartesian components of the vectors, we can write

A prime (‘) is often used to denote differentiation with respect to a scalar when the variable is clear from the context. In case of time a dot (.) is often used. Higher order derivatives and derivatives with respect to more than one scalar variable can be described in exactly in the same way as for ordinary functions and need no elaboration.

Since this definition of differentiation is formally the same as for derivative of a scalar function, usual rules of differential calculus apply in this case as well.

  • Notice that the order of the two vectors in equation (4) must be preserved; otherwise there is a sign reversal.

Sometimes it is more advantageous to write the result in terms of differentials rather than derivatives.

 

Example

2. Curves in space

Now

Then K is the curvature to the curve at that specific point and 1/K the radius of curvature.

 

3. Kinematics

where t is the time.  The velocity being the rate of change of position of the particle, is given by
The velocity is a vector in the direction of the tangent to the curve described by the particle. The magnitude v of the velocity is usually referred to as speed:

Similarly the acceleration is the rate of change of velocity, so

On using equation (27) in (29), we have
where is the curvature of the curve.  We have separated the acceleration of a particle moving along a curve into two components of which one is parallel to the tangent and the other is parallel to the curvature, that is, perpendicular to the tangent. The component of the acceleration parallel to the tangent is equal in magnitude to the rate of change of speed and is entirely independent of what sort of curve the particle is describing. On the other hand the component of the acceleration normal to the tangent is equal in magnitude to the product of the square of the speed of the particle and the curvature of the curve. But the rate of change of speed in its path has no effect at all on this normal component of the acceleration.

4. The gradient

 

We now come to vector differential operators, which appear when we consider fields. By fields we imply physical objects that depend on the position of the system or on the point under consideration. They may, in addition, also depend on the time variable. Relevant examples of fields are temperature in a room, electrostatic potential due to system of charges, gravitational potential due to a system of masses, charge density in a region of space and so on and so forth. All these are examples of scalar fields, where the physical object itself is a scalar quantity. Examples of vector fields are electric, magnetic or gravitational fields, momentum density in electromagnetic field and many more.

 

Let us first look at a typical scalar field V(x, y, z). We have used coordinates with respect to some Cartesian coordinate system to express the position of the point at which the scalar is being considered. If we consider the neighbouring point with coordinates (x+dx, y+dy, z+dz), the change in V is given by a theorem in partial differentiation:

Knowing the three partial derivatives along the three coordinates is enough to find variation of V along any direction.This relation can be written as the scalar product of two vectors:

4.1 Geometrical interpretation

The magnitude of the gradient vector gives the rate of increase of the scalar field under discussion.

 

Example

 

Find the gradient of r, the magnitude of the radius vector.

  • An alternative method of finding the gradient of a function is to make use of the equation (32):

 

Another example

 

Hence

4.2 Importance of operator 

 

The great importance of this operator in mathematical physics may be seen from a few illustrations.

where k is a constant depending upon the material of the body.

4.3 The operator  as a vector

 

The gradient of a scalar formally looks like a vector multiplying a scalar:

The term in the brackets is called del

This quantity has the appearance of a vector but is not a vector. In fact, by itself it has no meaning unless it “operates” on a function of coordinates. The operation is not one of multiplication but of differentiation. Hence  may be regarded as a vector operator, which on acting on a scalar field produces a vector quantity, the gradient. However, for all practical purposes this quantity may be regarded as a vector, it acts like a vector (and also a differential operator simultaneously) in all vector relations.

 

For example

4.4 The directional derivative

 

This is a scalar differential operator; when it operates on V(x, y, z), we get

 

5. The divergence, curl and Laplacian operators

 

5.1 The divergence

In the Cartesian coordinate system, the divergence of a vector takes the form

Like the dot product, divergence of a vector is a scalar quantity.

Consider the amount of fluid that passes through the faces of the cube parallel to the x-axis. The flux through the left hand face is

and through the right hand face is

Hence the net flux through faces parallel to the x-axis is

 

The total outward flux from the cube is therefore

where dV is the volume of the infinitesimal cube.

In case the fluid is incompressible, as much matter must leave the cube as enters it. The total change of contents must therefore be zero. For this reason the characteristic differential equation which any incompressible fluid must satisfy is

 This equation is often called the hydrodynamic equation. A vector whose divergence is zero is called solenoidal. The flow of an incompressible fluid is represented by a solenoidal vector.

 

Example-1

Find divergence of the position vector.

     Example-2

   

5.2 The curl

In Cartesian coordinates the curl of a vector can be written as


Example-1

Example-2

5.3 The product rules

 

Though some of these identities can be proved by direct vector methods, the simplest and the straight forward method is to appeal to tensor notation and the summation convention that will be introduced in a later module on tensors.

 

5.4 The Laplacian

 

Del is a first order differential operator, and consequently gradient, divergence and curl are first order derivatives. By applying this operator once again we can obtain second order derivatives. Various possibilities are:

Not all these give anything new. The first, gradient of divergence is just that-gradient of divergence. It does not occur very often in physics or engineering and has not been given any special name. The second, divergence of gradient, on expanding gives:

The third and the fourth second derivatives that we have written above are both zero:

Finally, it is easy to verify, especially by the tensor method that the fifth expression reduces to

This is just a combination of the first two terms, gradient of divergence and the Laplacian of a vector. Thus essentially we have only two second derivatives, of which one is seldom used. The only one of importance, therefore, is the Laplacian.

 

SUMMARY

 

  • We begin with a discussion of simple differentiation of a vector with respect to a scalar, like time.
  • Next we give a description of a curve in space and discuss the concept of curvature and radius of curvature.
  • Then we describe kinematics of the motion of a particle in vector form.
  • The most central concept of the derivative of a vector with respect to space coordinates, the gradient, is introduced. Geometrical interpretation of the gradient is also discussed.
  • Next we describe properties of the gradient, both as a differential operator and as a vector.
  • We then study the related differential operators of divergence and curl of a vector and the Laplacian operator.
  • Finally we write down the product rules when gradient, divergence and curl of more than one scalar or vector function is involved.