# Vector Calculus

**Line Integral***l*, the line integral is given by,**Surface Integral***S*, the surface integral or the flux of through*S*is defined as,**Volume Integral***F*over a volume*V*is written as,

# Vector Differentiations

In order to understand vector differentiation, we introduce an operator known as del operator known as

*del operator or differential vector operator*.**Differential Vector Operator (**âˆ‡**), or Del Operator**

The differential vector operator (âˆ‡), or

*del*or*nabla*, in Cartesian coordinates, is defined as,**Gradient of a Scalar****Definition**The gradient of a scalar function is both the magnitude and the direction of the maximum space rate of change of that function.**Mathematical Expression of Gradient**We consider a scalar function*F*. A mathematical expression for the gradient can be obtained by evaluating the difference in the field dF between the points*P*_{1}and*P*_{2}.

**Gradient of a scalar quantity**

Here,

*F*_{1}, and*F*_{2}are the contours on which*F*is constant.and, = Differential length = Differential displacement from

âˆ´

âˆ´ ; where,

*P*_{1}to*P*_{2}âˆ´

âˆ´ ; where,

*Î¸*is the angle between andFor maximum , i.e., when is in the direction of .

âˆ´ ; where is the normal derivative.

âˆ´ ; where is the normal derivative.

**Physical Interpretation**The gradient of a scalar quantity is the maximum space rate of change of the function.*T*, so at any point (*x, y, z*), the temperature is*T*(*x, y, z*), (assuming that the temperature does not change with time). Then, at any arbitrary point in the room, the gradient of*T*indicates the direction in which the temperature rises most rapidly. The magnitude of the gradient will determine how fast the temperature rises in that direction.

Similarly, if the scalar quantity is the electric potential

The relations of gradient in three different coordinate systems as,

*V*then grad*V*or âˆ‡*V*represents the potential gradient or electric field strength.The relations of gradient in three different coordinate systems as,

**Properties of Gradient**- The magnitude of the gradient of a scalar function is the maximum rate of change of the function per unit distance.
- The direction of the gradient of a scalar function is in the direction in which the function changes most rapidly.
- The gradient of a scalar function at any point is always perpendicular to the surface that passes through the point and over which the function is constant (points
*a*and*b*in the Figure). - The projection of the gradient of a scalar function (say, âˆ‡
*S*) in the direction of a unit vector i.e., , is known as the*directional derivative*of the function*S*along the unit vector

**Divergence of a Vector****Definition**Mathematically, divergence of a vector at any point is defined as the limit of its surface integral per unit volume as the volume enclosed by the surface around the point shrinks to zero.

where

*v*is the volume of an arbitrary-shaped region in space that includes the point,*S*is the surface of that volume, and the integral is a surface integral with being the outward normal to that surface. The result, div , is a function of the location of the point. From this definition, it also becomes explicitly visible that div can be seen as the source density of the flux of**Mathematical Expression of Divergence**We consider a hypothetical infinitesimal cubical box oriented along the coordinate axes around an infinitesimal region of space.*P*(*x, y, z*). Let,*V*_{1},*V*_{2}, and*V*_{3}be the components of along the three coordinate axes.

**To derive expression for divergence in cartesian coordinates**

â€‹The component vectors are given as follows.

at back surface,

*Along x-direction*: at front surface,at back surface,

*Along y-direction*: at left surface,

at back surface,

*Along z-direction*: at top surface,

â€‹at bottom surface,

Therefore, the net outward flux of the vector is,

*Along x-direction*:

*Along y-direction*:

*Along z-direction*:

The total net outward flow, considering all three directions, is,

â€‹

where is the infinitesimal volume of the cube.

Hence, the total net outward flow per unit volume is given as,

By definition, this is the divergence of the vector.

**Physical Interpretation**The physical significance of the divergence of a vector field is the rate at which the density of a vector exits a given region of space. The definition of the divergence, therefore, complies with the natural fact that in the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into or out of the region.**Properties of Divergence**- The result of the divergence of a vector field is a scalar.
- Divergence of a scalar field has no meaning.
- Divergence may be positive, negative or zero. A vector field with constant
*zero divergence*is called*solenoidal*; in this case, no net flow can occur across any closed surface.

**Curl of a Vector****Definition**The curl of a vector field, denoted as curl or is defined as the vector field having magnitude equal to the maximum circulation at each point and oriented perpendicularly to this plane of circulation for each point.

Mathematically, it is defined as the limit of the ratio of the integral of the cross product of the vector with outward drawn normal over a closed surface, to the volume enclosed by the surface, as the volume tends to zero.

In other words, the component of the curl of a vector in the direction of the unit vector is the ratio of the line integral of the vector around a closed contour, to the area enclosed by the contour, as the area tends to zero.

**Mathematical Expression of Curl**In order to find an expression for the curl of a vector, we consider an elemental area in the*yz*-plane as shown in the figure.

**To derive expression for curl in Cartesian coordinates**

We define a vector at the centre of the area

*P*(*x, y, z*).**Physical Interpretation**The physical significance of the curl of a vector at any point is that it provides a measure of the amount of rotation or angular momentum of the vector around the point.*xy*-plane.

â€‹

**Rotation of a floating leaf and interpretation of curl**

If the velocity at the surface is only in the

*y*-direction and is uniform over the surface, there will be no circulation of the leaf.

â€‹

**Interpretation of positive and negative velocity gradients**

In this case, the rotation is about the

*z*-axis and the curl of the velocity vector in the z-direction is written as . A positive value of implies a rotation from

*x*to

*y*, i.e., anticlockwise.

It is seen that

- For positive value of , rotation is anticlockwise.
- For negative value of , rotation is clockwise.

**Properties of Curl**- The result of the curl of a vector field is another vector field.
- Curl of a scalar field has no meaning.
- If the value of the curl of a vector field is zero then the vector field is said to be an
*irrotational*or*conservative*field. An electrostatic field is one such field.

**Laplacian (âˆ‡**^{2}) Operator

The Laplacian operator (âˆ‡
Practically, it is a single operator which is the composite of gradient and divergence operators.
The Laplacian of a scalar field is also a scalar field.
The Laplacian of a scalar field

^{2}) can operate both on scalar as well as vector fields.**Laplacian (âˆ‡**^{2}*The Laplacian operator (âˆ‡***) of a Scalar**^{2}) of a scalar field is the divergence of the gradient of the scalar field upon which the operator operates.*F*in Cartesian coordinates is written as,