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In multivariable calculus, the directional derivative measures the rate at which a function changes in a particular direction at a given point. [citation needed]The directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a direction ...
The gradient is related to the differential by the formula = for any , where is the dot product: taking the dot product of a vector with the gradient is the same as taking the directional derivative along the vector.
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.
In Cartesian coordinates, the divergence of a continuously differentiable vector field = + + is the scalar-valued function: = = (, , ) (, , ) = + +.. As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.
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:
From this follows that the directional derivative is the inner product of its direction by the vector derivative. All needs to be observed is that the direction a {\displaystyle a} can be written a = ( a ⋅ e i ) e i {\displaystyle a=(a\cdot e^{i})e_{i}} , so that:
Geometrically, the derivative at a point is the slope of the tangent line to the graph of the function at that point, provided that the derivative exists and is defined at that point. For a real-valued function of a single real variable, the derivative of a function at a point generally determines the best linear approximation to the function ...
A formula to determine functional derivatives for a common class of functionals can be written as the integral of a function and its derivatives. This is a generalization of the Euler–Lagrange equation : indeed, the functional derivative was introduced in physics within the derivation of the Lagrange equation of the second kind from the ...