Source : ct|01.06.13

< Mathematics and physics

Gradient of a scalar field

"The equations that contain some differentials (either ordinary or partial) show, as we know, the relationship between the variables involved in these equations, and the derivates that represent the relative infinitely small increases that they take when they varies in accordance with their mutual dependence (dependance that the nature of the question that we will solve establishes between them)."
André-Marie Ampère (1775-1836) - General considerations on the integrals of partial differential equations (1814)

The dictionary says that "the gradient is the rate of change of a meteorological element as a function of distance". In mathematics and physics, we talk about gradient of a scalar field (or gradient of a scalar potential ) . What is the precise definition of this concept and what is it exactly? ...

1) Definition

Considering a scalar field U(x,y,z)
We call gradient of U the vector

we also note this vector

with i =(1,0,0), j=(0,1,0), k=(0,0,1), and the nabla operator equal to

2) Meaning

To illustrate concretely at a point M(x,y,z), the vector V(x,y,z) = grad U(x,y,z) of a scalar field U(x,y,z), we examine the simple case of a scalar field U(x) (with one dimension) or U(x,y) (with two dimensions).

• With one dimension, the vector V=grad U(x) of a scalar field U(x) at a point M(x) defines the slope (tan) of this field U(x) in this point.

dU/dx is the derivative of U(x) at the point M(x) and is the slope of the tangent to the curve U(x) in this point. It represents the infinitesimal variation of the function with respect to an infinitesimal movement around this point.

• With two dimensions, the components of the vector V=grad U(x,y) of the scalar field U(x,y) at a point M(x,y) represent the infinitesimal variations of this field in the x and y directions relative to an infinitesimal movement in these directions. The vector V=grad U(x,y) defines the slope (direction of greatest variation) of the field U (x, y) at this point.

Generalisation

More generally, we consider an infinitely small path dr = dx i + dy j +dz k in a space (0,x,y,z) containing a scalar field U U(x,y,z). The circulation of the vector V=grad U along this path is equal to

Thus the circulation of the vector gradient U between two points A and B of any path (AB) is equal to

The circulation (between two points) of the gradient of a scalar field (or gradient of a scalar potential) is equal to the difference between the values of this field between these two points (potential difference).

Example

Verify the formula

in the particular case U(x,y)=x.y

Answer
dU = U(x+dx,y+dy)-U(x,y)= (x+dx)(y+dy)-xy = xdy + ydx + dxdy avec xdy + ydx + dxdy which id equal to xdy + ydx car, because dx and dy being infinitely small, dxdy is negligible relative to xdy and ydx.

Gradient in cylindrical coordinates

Consider, in cylindrical coordinates, a scalar field U(r,θ,z) and a vector E=grad U.
E = Er u + Eθ v + Ez k
dr = dr u + rdθ v + dz k

dU = grad U. dr = Er.dr + Eθ.rdθ + Ez.dz
hence

Gradient in spherical coordinates

Consider, in spherical coordinates, a scalar field U(r,θ,φ) and a vector E=grad U.
E = Er u + Eθ v + Eφ w
dr = dr u + rdθ v + rsindφ w

dU = grad U.dr = Er.dr + Eθ.rdθ + Eφ.rsinθdφ
hence