In vector calculus, 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. More rigorously, the gradient of a function from the Euclidean space Rn to R is the best linear approximation to that function at any particular point in Rn. To that extent, the gradient is a particular case of the Jacobian.
The word gradient is sometimes used synonymously with grade, meaning the inclination of a surface along a given direction. One can obtain the grade by taking the dot product of the vector gradient with the unit vector in the direction of interest. The magnitude of the gradient is also sometimes referred to as just the gradient.
Interpretations of the gradient
Consider a room in which the temperature is given by a scalar field φ, so at each point (x,y,z) the temperature is φ(x,y,z). 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 it gets hot most quickly. The magnitude of the gradient will tell how fast it gets hot in that direction.
Consider a hill whose height at a point (x,y) is H(x,y). The gradient of H 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 H 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 of a scalar function Template:Phisymbol is denoted by:
In 3 dimensions, the expression expands to
Although expressed in terms of coordinates, the result is invariant under orthogonal transformations, as it should, in view of the geometric definition.
The gradient of the function Template:Phisymbol = 2x + 3y2 − sin(z) is:
The gradient on manifolds
where denotes the inner product on M (the metric) and ξf is the function that takes any point p to the directional derivative of f in the direction ξ evaluated at p. In other words, under some coordinate chart, ξf(p) will be:
The gradient of a function is related to the exterior derivative, since ξf(p) = df(ξ). 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.
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