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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.
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions f and g in terms of the derivatives of f and g.More precisely, if = is the function such that () = (()) for every x, then the chain rule is, in Lagrange's notation, ′ = ′ (()) ′ (). or, equivalently, ′ = ′ = (′) ′.
If f is a function, then its derivative evaluated at x is written ′ (). It first appeared in print in 1749. [3] Higher derivatives are indicated using additional prime marks, as in ″ for the second derivative and ‴ for the third derivative. The use of repeated prime marks eventually becomes unwieldy.
With addition and scalar multiplication defined as this, F X is a vector space over F. Now, fix two elements a, b in X and define a function e from X to F by e(x) = 1 F for all x in X. Define L 1 and L 2 from F X to F by L 1 (f) = f(a) and L 2 (f) = f(b). Then L 1 and L 2 are two linear functionals on F X such that L 1 (e)= L 2 (e)= 1 F For f ...
The tensor derivative of a vector field (in three dimensions) is a 9-term second-rank tensor – that is, a 3×3 matrix – but can be denoted simply as , where represents the dyadic product. This quantity is equivalent to the transpose of the Jacobian matrix of the vector field with respect to space.
In calculus, the inverse function rule is a formula that expresses the derivative of the inverse of a bijective and differentiable function f in terms of the derivative of f. More precisely, if the inverse of f {\displaystyle f} is denoted as f − 1 {\displaystyle f^{-1}} , where f − 1 ( y ) = x {\displaystyle f^{-1}(y)=x} if and only if f ...
However a norm-coercive mapping f : R n → R n is not necessarily a coercive vector field. For instance the rotation f : R 2 → R 2 , f ( x ) = (− x 2 , x 1 ) by 90° is a norm-coercive mapping which fails to be a coercive vector field since f ( x ) ⋅ x = 0 {\displaystyle f(x)\cdot x=0} for every x ∈ R 2 {\displaystyle x\in \mathbb {R ...
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 ...