Vecctor analysis and got the mathematical formulae for gradient

In summary: The divergence of a vector field is asociated with conserved quantities, if the divergence is zero there are no "sources" or "sinks". Divergence is a measure of how far away two points are from each other along a vector field, and is always positive. Curl is a measure of how much a vector field has rotated around its origin, and is always negative.
  • #1
hershal
10
0
I was reading vecctor analysis and got the mathematical formulae for gradient but could not understand its physical meaning.
What is the physical meaning of gradient of a scalar ? And of a vector .
Also, I wanted to know the physical meanings of Divergence and Curl .
 
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  • #2
I'd suggest a book reference - "Div, Grad, Curl and all that".



The physical meaning of the gradient of a scalar function is that it's the steepness of the slope. Imagine height being a scalar, then the gradient of the height would be a vector pointing "uphill", the length of the vector is proportional to the steepness of the slope - in civil engineering turns the gradient (note the similarity) of a road running directly uphill/downhill.

Divergence of a vector field is asociated with conserved quantities, if the divergence is zero there are no "sources" or "sinks".

Curl of a vector field is associated with it's rotation, if the curl is zero the field is irrotational.

This may not be detailed enough - it's a tricky subject, but the book I quoted is really very good at providing detailed examples and physical explanations.
 
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  • #3
Thanks ! pervect, i'll see if i can get that grad,div,curl book .
 
  • #4
The gradient is a differential operator on a scalar field, [tex]\phi[/tex]. The gradient, grad[tex]\phi[/tex], is a "vector field" defined by the requirement that

grad[tex]\,\phi\,\cdot[/tex] ds = d[tex]\,\phi[/tex]

where d[tex]\,\phi[/tex] is the differential change in the scalar field, [tex]\phi[/tex], corresponding to the arbitrary space displacement, ds, and from this,

d[tex]\,\phi[/tex] = | grad [tex]\,\phi\,[/tex]| |ds| cos [tex]\theta[/tex], where is the angle between the displacement vector and the line formed between two points of interest in the scalar field.

Since cos [tex]\theta[/tex] has a maximum value of 1, that is when [tex]\theta[/tex]=0, it is clear that the rate of change is greatest if the differential displacement is in the direction of grad[tex]\,\phi\,[/tex], or stated another way,

"The direction of the vector grad[tex]\,\phi[/tex] is the direction of maximum rate of change (spatially-speaking) of [tex]\,\phi[/tex] from the point of consideration, i.e. direction in which [tex]\frac{d\phi}{ds}[/tex] is greatest."

The gradient of [tex]\phi[/tex] is considered 'directional derivative' in the direction of the maximum rate of change of the scalar field [tex]\phi[/tex].

Think of contours of elevation on a mountain slope. Points of the same (constant) elevation have the same gravitational potential, [tex]\phi[/tex]. Displacement along (parallel) to the contours produce no change in [tex]\phi[/tex] (i.e. d[tex]\phi[/tex] = 0). Displacements perpendicular (normal) to the equipotential are oriented in the direction of most rapid change of altitude, and d[tex]\phi[/tex] has the maximum value.

Isotherms are equipotentials with respect to heat flow.

See related discussion on the directional derivative (forthcoming).

Examples of scalar fields:

  • temperature
  • density (mass distribution) in an object or matter (solid, liquid, gas, . . .)
  • electrostatic (charge distribution)

Examples of vector fields:

  • gravitational force
  • velocity at each point in a moving fluid (e.g. hurricane or tornado, river, . . .)
  • magnetic field intensity

I am doing something similar for div and curl
 
  • #5
Thanks Astronuc, I can't wait.
 

1. What is vector analysis?

Vector analysis is a branch of mathematics that deals with the study of vectors, which are mathematical quantities that have both magnitude and direction. It involves the use of various mathematical operations, such as addition, subtraction, and multiplication, to manipulate and analyze vectors.

2. What is the gradient of a vector?

The gradient of a vector is a mathematical operation that results in a vector that represents the direction and magnitude of the steepest slope of a function at a given point. It is often used in vector calculus to calculate rates of change and to find the direction of maximum increase of a function.

3. How do you calculate the gradient of a vector?

The gradient of a vector can be calculated using the partial derivative of each component of the vector with respect to each coordinate axis. For example, if we have a vector v = (x, y, z), the gradient of v is given by ∇v = (∂v/∂x, ∂v/∂y, ∂v/∂z).

4. What is the relationship between gradient and direction?

The gradient of a vector represents the direction of maximum increase of a function at a given point. This means that the gradient vector points in the direction of the steepest slope of the function. In other words, the direction of the gradient vector is perpendicular to the level curves of the function.

5. How is gradient used in real-life applications?

Gradient is used in various fields, such as physics, engineering, and economics, to analyze and solve problems involving rates of change, optimization, and direction of movement. It is also used in computer graphics to create realistic lighting and shading effects. Additionally, gradient descent algorithms, which use the gradient to find the minimum of a function, are widely used in machine learning and artificial intelligence.

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