Gradient
A generalization of these concepts is the gradient in vector calculus; and this article is mostly about this vector gradient. The
gradient of a scalar field is a
vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.
A generalization of the gradient, for functions which have vectorial values, is the Jacobian.
Encyclopedia
- See slope for the measure of steepness of a straight line.
- See grade for the grade or gradient of roads and other geographic features.
A generalization of these concepts is the
gradient in vector calculus; and this article is mostly about this vector gradient. The
gradient of a scalar field is a
vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.
A generalization of the gradient, for functions which have vectorial values, is the Jacobian.
Interpretations of the gradient
Consider a room in which the temperature is given by a scalar field , so at each point the temperature is . We will assume that the temperature does not change in time. Then, at each point in the room, the gradient at that point will show the direction in which the temperature rises most quickly. The magnitude of the gradient will tell how fast the temperature rises in that direction.
Consider a hill whose height at a point is . The gradient of at a point is in the direction of the steepest
slope/grade at that point. The magnitude of the gradient tells how steep the slope actually is.
The gradient can also be used to tell how things change in other directions rather than the direction of largest change. Consider again the example with the hill. One can have a road which goes right uphill where the slope is largest and then its slope is the magnitude of the gradient. Or one can have a road which goes under an angle with the uphill direction, say for example an angle of 60° when projected onto the horizontal plane. Then, if the steepest slope on the hill is 40%, the road will make a shallower slope of 20% which is 40% times the cosine of 60°.
This observation can be mathematically stated as follows. The gradient of the hill height function
dotted with a unit vector gives the slope of the surface in the direction of the vector. This is called the directional derivative.
The gradient is irrotational and thus line integrals through a gradient field are path independent and can be evaluated with the gradient theorem.
Formal definition
The gradient of a scalar function with respect to a vector variable is denoted by where denotes the vector differential operator del.
These other symbols are equivalent and carry the same meaning: , .
By definition, the gradient is a column vector whose components are the partial derivatives of . That is:
-
Although expressed in terms of coordinates, the result is invariant under orthogonal transformations, as it should, in view of the geometric definition.
Example
In 3
dimensions, the expression expands to in
Cartesian coordinates.
For example, the gradient of the function
-
is:
Linear approximation to a function
The gradient of a function from the Euclidean space
Rn to
R characterizes the best
linear approximation to that function at any particular point in
Rn.
The approximation is as follows:
-
for close to , where is the gradient computed at .
The gradient on manifolds
For any differentiable function f on a Riemannian manifold
M, the gradient of
f is the
vector field such that for any vector ,
where denotes the
inner product on
M and
is the function that takes any point
p to the directional derivative of in the direction evaluated at
p. In other words, under some coordinate chart, will be:
The gradient of a function is related to the
exterior derivative, since . Indeed, the metric allows one to associate canonically the 1-form
df to the vector field . In
Rn the flat metric is implicit and the gradient can be identified with the exterior derivative.
See also
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