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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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- #1

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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

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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.

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 .

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Astronuc

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

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

d[tex]\,\phi[/tex] = | grad [tex]\,\phi\,[/tex]| |d

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

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Thanks Astronuc, I can't wait.

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