# Vecctor analysis and got the mathematical formulae for gradient

 Admin P: 21,869 Vecctor analysis and got the mathematical formulae for gradient 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