Gradient Totally Explained

Everything about Gradient totally explained

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.
A generalization of the gradient, for functions on a Banach space which have vectorial values, is the Jacobian.

Interpretations of the gradient

Consider a room in which the temperature is given by a scalar field

T

, so at each point

(x,y,z)

the temperature is

T(x,y,z)

(we will assume that the temperature doesn't change in time). Then, at each point in the room, the gradient of

T

at that point will show the direction in which the temperature rises most quickly. The magnitude of the gradient will determine how fast the temperature rises in that direction.
Consider a hill whose height above sea level at a point

(x, y)

H(x, y)

. The gradient of

H

at a point is a vector pointing in the direction of the steepest slope or grade at that point. The steepness of the slope at that point is given by the magnitude of the gradient vector.
The gradient can also be used to measure how a scalar field changes in other directions, rather than just the direction of greatest change, by taking a dot product. Consider again the example with the hill and suppose that the steepest slope on the hill is 40%. If a road goes directly up the hill, then the steepest slope on the road will also be 40%. If instead, the road goes around the hill at an angle with the uphill direction (the gradient vector), then it'll have a shallower slope. For example, if the angle between the road and the uphill direction, projected onto the horizontal plane, is 60°, then the steepest slope along the road will be 20% which is 40% times the cosine of 60°.
This observation can be mathematically stated as follows. If the hill height function

H

is differentiable, then the gradient of

H

dotted with a unit vector gives the slope of the hill in the direction of the vector. More precisely, when

H

is differentiable the dot product of the gradient of H with a given unit vector is equal to the directional derivative of H in the direction of that unit vector.

Formal definition

The gradient (or gradient vector field) of a scalar function

f(x)

with respect to a vector variable

x = (x_1,dots,x_n)

is denoted by

abla f

vec.

Generalizing the case M=Rⁿ, the gradient of a function is related to its exterior derivative, since

(partial_X f) (x) = df_x(X_x)

. More precisely, the gradient

abla f

is the vector field associated to the differential 1-form df using the musical isomorphism

sharp=sharp^gcolon T^*M	o TM

(called "sharp") defined by the metric g. The relation between the exterior derivative and the gradient of a function on Rⁿ is a special case of this in which the metric is the flat metric given by the dot product.

Further Information

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