Vecctor analysis and got the mathematical formulae for gradient

[hershal]

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 .

 

[pervect]

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.

 

[hershal]

Thanks ! pervect, i'll see if i can get that grad,div,curl book .

 

[Astronuc]

The gradient is a differential operator on a scalar field, \phi. The gradient, grad\phi, is a "vector field" defined by the requirement that

grad\,\phi\,\cdot ds = d\,\phi

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

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

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

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

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

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

 

[KingOfTwilight]

Thanks Astronuc, I can't wait.