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Vector Calculus

  1. Line Integral
    The line integral of a vector is the integral of the dot product of the vector and the differential-length vector tangential to a specified path.
    For the vector Description: 78239.png and a path l, the line integral is given by,
    Description: 78230.png
  2. Surface Integral
    For a vector Description: 78087.png continuous in a region containing a smooth surface S, the surface integral or the flux of Description: 78076.png through S is defined as,
    Description: 78069.png
  3. Volume Integral
    The volume integral of a scalar quantity F over a volume V is written as,
    Description: 77977.png
    The concept of volume integrals is necessary to calculate the charge or mass of an object which are distributed in the volume.

Vector Differentiations

In order to understand vector differentiation, we introduce an operator known as del operator known as del operator or differential vector operator.
  1. Differential Vector Operator (), or Del Operator
The differential vector operator (∇), or del or nabla, in Cartesian coordinates, is defined as,
 Description: 77579.png
  1. Gradient of a Scalar
    1. Definition The gradient of a scalar function is both the magnitude and the direction of the maximum space rate of change of that function.
    2. 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 P1 and P2.

Gradient of a scalar quantity

Here, F1, and F2 are the contours on which F is constant.
Description: 77495.png
where, Description: 77488.png
and, Description: 77480.png = Differential length = Differential displacement from P1 to P2
∴ Description: 77461.png ; where, θ is the angle between Description: 77453.png and Description: 77449.png
For maximum Description: 77441.pngDescription: 77433.png i.e., when Description: 77426.png is in the direction of Description: 77417.png.
∴  Description: 77407.png; where Description: 77400.png is the normal derivative.
Thus, by definition of gradient, we have the mathematical expression of gradient in Cartesian coordinates given as,
  Description: 77392.png
  1. Physical Interpretation The gradient of a scalar quantity is the maximum space rate of change of the function.
    For example, we consider a room in which the temperature is given by a scalar field 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 V then grad V or V represents the potential gradient or electric field strength.

The relations of gradient in three different coordinate systems as,
Description: 77362.png
  1. Properties of Gradient
    1. The magnitude of the gradient of a scalar function is the maximum rate of change of the function per unit distance.
    2. The direction of the gradient of a scalar function is in the direction in which the function changes most rapidly.
    3. 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).
    4. The projection of the gradient of a scalar function (say, S) in the direction of a unit vector Description: 77354.png i.e., Description: 77347.png, is known as the directional derivative of the function S along the unit vector Description: 77339.png
  1. Divergence of a Vector
    1. 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.
Description: 77130.png
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 Description: 77123.png being the outward normal to that surface. The result, div Description: 77116.pngis a function of the location of the point. From this definition, it also becomes explicitly visible that div Description: 77107.png can be seen as the source density of the flux of Description: 77100.png
  1. Mathematical Expression of Divergence We consider a hypothetical infinitesimal cubical box oriented along the coordinate axes around an infinitesimal region of space.
    We consider a vector Description: 102660.png at a point P(x, y, z). Let, V1V2, and V3 be the components of Description: 102671.png along the three coordinate axes.
    In order to compute the surface integral, we see that, there are six surfaces to this box, and the net content leaving the box is therefore simply the sum of differences in the values of the vector field along the three sets of parallel surfaces of the box.

To derive expression for divergence in cartesian coordinates
The component vectors are given as follows.
Along x-direction: at front surface, Description: 77044.png
at back surface, Description: 77035.png
Along y-direction: at left surface, Description: 77027.png
at back surface, Description: 77019.png
Along z-direction: at top surface, Description: 77011.png
at bottom surface, Description: 77004.png 

Therefore, the net outward flux of the vector is,
Along x-directionDescription: 76997.png
Along y-directionDescription: 76988.png
Along z-directionDescription: 76980.png

The total net outward flow, considering all three directions, is,
Description: 76971.png 
where Description: 76962.png is the infinitesimal volume of the cube.

Hence, the total net outward flow per unit volume is given as,
Description: 76952.png

By definition, this is the divergence of the vector.

 Description: 76943.png
  1. 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.
    In other words, divergence of a vector field at a given point is an operator that measures the magnitude of the source or sink, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
    For example, we consider air as it is heated or cooled. The relevant vector field for this example is the velocity of the moving air at a point. If air is heated in a region, it will expand in all directions such that the velocity field points outward from that region. Therefore, the divergence of the velocity field in that region would have a positive value, as the region is a source. If the air cools and contracts, the divergence is negative and the region is called a sink.
  2. Properties of Divergence
    1. The result of the divergence of a vector field is a scalar.
    2. Divergence of a scalar field has no meaning.
    3. 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.
      For example, for an incompressible fluid, if Description: 76813.png denotes the quantity of the fluid at any point then Description: 76805.png, i.e., an incompressible fluid cannot diverge from, nor converge towards a point.
  3. Curl of a Vector
    1. Definition The curl of a vector field, denoted as curl Description: 76623.png or Description: 76613.png 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.
Description: 76603.png
In other words, the component of the curl of a vector in the direction of the unit vector Description: 76595.png 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.
Description: 76588.png
where the direction of the contour is obtained from right-hand corkscrew rule.
  1. 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 Description: 76578.png at the centre of the area P(x, y, z).
The closed line integral of Description: 76571.png around the path abcd is,
Description: 76564.png
  1. 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.
    We consider a stream on the surface of which floats a leaf, in the xy-plane.
Description: 76504.png
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.
But, if there are vertices or eddies in the stream, there will be rotational movement of the leaf.
The rate of rotation or angular velocity at any point is a measure of the curl of the velocity of the stream at that point.
Description: 76465.png
Interpretation of positive and negative velocity gradients

In this case, the rotation is about the z-axis and the curl of the velocity vector Description: 100391.png in the z-direction is written as Description: 100385.png. A positive value of Description: 100377.png implies a rotation from x to y, i.e., anticlockwise.

It is seen that
  • For positive value of Description: 100210.png, rotation is anticlockwise.
  • For negative value of Description: 76457.png, rotation is clockwise.
  1. Properties of Curl
    1. The result of the curl of a vector field is another vector field.
    2. Curl of a scalar field has no meaning.
    3. 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.
  1. Laplacian (2) Operator
The Laplacian operator (2) can operate both on scalar as well as vector fields.
Laplacian (2) of a Scalar The Laplacian operator (2) of a scalar field is the divergence of the gradient of the scalar field upon which the operator operates.
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 F in Cartesian coordinates is written as,

Description: 76158.png

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