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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 ...
This article uses the standard notation ISO 80000-2, which supersedes ISO 31-11, for spherical coordinates (other sources may reverse the definitions of θ and φ): . The polar angle is denoted by [,]: it is the angle between the z-axis and the radial vector connecting the origin to the point in question.
The directional derivative of a scalar field (,,) in the direction (,,) = ^ + ^ + ^ ... The 3 remaining vector derivatives are related by the equation:
Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term ...
In vector calculus the derivative of a vector y with respect to a scalar x is known as the tangent vector of the vector y, . Notice here that y : R 1 → R m . Example Simple examples of this include the velocity vector in Euclidean space , which is the tangent vector of the position vector (considered as a function of time).
If all the partial derivatives of exist and are continuous at , then they determine the directional derivative of in the direction by the formula: [45] = =. Total derivative, total differential and Jacobian matrix
The directional derivative of with respect to in the direction , where and are multivectors, is defined as = (+) , ... This equation is just expressing ...
The derivatives in the table above are for when the range of the inverse secant is [,] and when the range of the inverse cosecant is [,]. It is common to additionally define an inverse tangent function with two arguments , arctan ( y , x ) {\textstyle \arctan(y,x)} .