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One way to define the curl of a vector field at a point is implicitly through its components along various axes passing through the point: if ^ is any unit vector, the component of the curl of F along the direction ^ may be defined to be the limiting value of a closed line integral in a plane perpendicular to ^ divided by the area enclosed, as ...
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 ...
This extends the directional derivative of scalar functions to sections of vector bundles or principal bundles. In Riemannian geometry, the existence of a metric chooses a unique preferred torsion-free covariant derivative, known as the Levi-Civita connection. See also gauge covariant derivative for a treatment oriented to physics.
An alternative definition: A smooth vector field on a manifold is a linear map : () such that is a derivation: () = + for all , (). [ 3 ] If the manifold M {\displaystyle M} is smooth or analytic —that is, the change of coordinates is smooth (analytic)—then one can make sense of the notion of smooth (analytic) vector fields.
The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics.These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical simulations.
The ratio in the definition of the derivative is the slope of the line through two points on the graph of the function , specifically the points (, ()) and (+, (+)). As h {\displaystyle h} is made smaller, these points grow closer together, and the slope of this line approaches the limiting value, the slope of the tangent to the graph of ...
The directional derivative is linear regarding its direction, that is: ∇ α a + β b = α ∇ a + β ∇ b . {\displaystyle \nabla _{\alpha a+\beta b}=\alpha \nabla _{a}+\beta \nabla _{b}.} From this follows that the directional derivative is the inner product of its direction by the vector derivative.
Gradient of the 2D function f(x, y) = xe −(x 2 + y 2) is plotted as arrows over the pseudocolor plot of the function.. Consider a room where the temperature is given by a scalar field, T, so at each point (x, y, z) the temperature is T(x, y, z), independent of time.