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.