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Vecctor analysis and got the mathematical formulae for gradient 
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#1
Aug3005, 12:16 PM

P: 10

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 . 


#2
Aug3005, 12:25 PM

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Sci Advisor
P: 7,596

I'd suggest a book reference  "Div, Grad, Curl and all that".
http://www.amazon.com/exec/obidos/tg...08712?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. 


#3
Sep105, 09:30 AM

P: 10

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



#4
Sep105, 10:26 AM

Admin
P: 21,808

Vecctor analysis and got the mathematical formulae for gradient
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 (spatiallyspeaking) 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:
Examples of vector fields:
I am doing something similar for div and curl 


#5
Sep105, 11:59 AM

P: 37

Thanks Astronuc, I can't wait.



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