Gradient, Divergence and Curl
| Site: | Learn@GC Tanur |
| Course: | 2022-ELE5B10-Electromagnetic Theory |
| Book: | Gradient, Divergence and Curl |
| Printed by: | Guest user |
| Date: | Thursday, 19 September 2024, 6:56 AM |
1. Del Operator
Del, or nabla, is an vector differential operator, usually represented by the nabla symbol ∇.
The del symbol can be interpreted as a vector of partial derivative operators, and its three possible meanings—gradient, divergence, and curl—can be formally viewed as the product with a scalar, a dot product, and a cross product, respectively, of the del "operator" with the field.
These formal products do not necessarily commute with other operators or products. These three uses, detailed below, are summarized as:
- Gradient:
- Divergence:
- Curl:
2. Gradient of a Vector
The gradient of a scalar field is a vector field and whose magnitude is the rate of change and which points in the direction of the greatest rate of increase of the scalar field.
The gradient always points in the direction of maximum change or the steepest ascent.
And for a three-dimensional scalar field ∅ (x, y, z)
The gradient of a scalar field is the derivative of f in each direction. Note that the gradient of a scalar field is a vector field. An alternative notation is to use the del or nabla operator, ∇f = grad f.
For a three dimensional scalar, its gradient is given by:
Gradient is a vector that represents both the magnitude and the direction of the maximum space rate of increase of a scalar.
dV = (∇V) ∙ dl, where dl = ai ∙ dl
In Cartesian
In Cylindrical
In Spherical
Properties of gradient
· We can change the vector field into a scalar field only if the given vector is differential. The given vector must be differential to apply the gradient phenomenon.
· The gradient of any scalar field shows its rate and direction of change in space.
Example 1: For the scalar field ∅ (x,y) = 3x + 5y,calculate gradient of ∅.
Solution 1: Given scalar field ∅ (x,y) = 3x + 5y
Example 2: For the scalar field ∅ (x,y) = x4yz,calculate gradient of ∅.
Solution: Given scalar field ∅ (x,y) = x4yz
Example 3: For the scalar field ∅ (x,y) = x2sin5y,calculate gradient of∅.
Solution: Given scalar field ∅ (x,y) = x2sin5y
3. Divergence Video
Topic : Divergence of a vector.
Part 2
4. Divergence
Divergence is an operation on a vector field that tells us how the field behaves toward or away from a point.
Divergence of a vector field at a particular point P is a measure of the “outflowingness” of the vector field at P .
If F⇀ represents the velocity of a fluid, then the divergence of F⇀ at P measures the net rate of change with respect to time of the amount of fluid flowing away from P (the tendency of the fluid to flow “out of” P). In particular, if the amount of fluid flowing into P is the same as the amount flowing out, then the divergence at P is zero.