Vecctor analysis and got the mathematical formulae for gradient

  1. 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. jcsd
  3. pervect

    pervect 8,158
    Staff Emeritus
    Science Advisor

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

    Staff: Mentor

    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
  6. Thanks Astronuc, I can't wait.
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