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For example, if the road is at a 60° angle from the uphill direction (when both directions are projected onto the horizontal plane), then the slope along the road will be the dot product between the gradient vector and a unit vector along the road, as the dot product measures how much the unit vector along the road aligns with the steepest ...
The gradient theorem states that if the vector field F is the gradient of some scalar-valued function (i.e., if F is conservative), then F is a path-independent vector field (i.e., the integral of F over some piecewise-differentiable curve is dependent only on end points). This theorem has a powerful converse:
If W is a vector field with curl(W) = V, then adding any gradient vector field grad(f) to W will result in another vector field W + grad(f) such that curl(W + grad(f)) = V as well. This can be summarized by saying that the inverse curl of a three-dimensional vector field can be obtained up to an unknown irrotational field with the Biot–Savart ...
A vector field V defined on an open set S is called a gradient field or a conservative field if there exists a real-valued function (a scalar field) f on S such that = = (,,, …,). The associated flow is called the gradient flow , and is used in the method of gradient descent .
In vector calculus, a conservative vector field is a vector field that is the gradient of some function. [1] A conservative vector field has the property that its line integral is path independent; the choice of path between two points does not change the value of the line integral. Path independence of the line integral is equivalent to the ...
The dotted vector, in this case B, is differentiated, while the (undotted) A is held constant. The utility of the Feynman subscript notation lies in its use in the derivation of vector and tensor derivative identities, as in the following example which uses the algebraic identity C⋅(A×B) = (C×A)⋅B:
In vector calculus, the del operator is used to define the concepts of gradient, divergence, and curl in terms of partial derivatives. A matrix of partial derivatives, the Jacobian matrix, may be used to represent the derivative of a function between two spaces of arbitrary dimension.
By example, in physics, the electric field is the negative vector gradient of the electric potential. The directional derivative of a scalar function f(x) of the space vector x in the direction of the unit vector u (represented in this case as a column vector) is defined using the gradient as follows.