Vecctor analysis and got the mathematical formulae for gradient

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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
pervect
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I'd suggest a book reference - "Div, Grad, Curl and all that".

https://www.amazon.com/exec/obidos/tg/detail/-/0393969975/104-4551598-1508712?v=glance

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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Thanks ! pervect, i'll see if i can get that grad,div,curl book .
 
  • #4
Astronuc
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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.
 

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