computes the directional derivative of a scalar field in the direction given by a vector
the scalar or vector field to differentiate
Vector(algebraic); the direction Vector or vector field
point=list(algebraic) or point=Vector(algebraic); point where the derivative will be evaluated
list(algebraic) or Vector(algebraic); components specifying the direction of the directional derivative in a specified coordinate system
(optional) list(name) or symbol[name, name, ...]; list of names or name of the coordinate system indexed by the coordinate names
The DirectionalDiff(F,v,c) command, where F is a scalar function, computes the directional derivative of F at the location and direction specified by v. The expression F is interpreted in the coordinate system specified by c, if provided, and otherwise in the current coordinate system.
The DirectionalDiff(F,v,c) command, where F is a VectorField, computes the VectorField of directional derivatives of each component of F with respect to v.
The argument v can be a free Vector in Cartesian coordinates, a position Vector, a vector field or a rooted Vector. If v is one of the first three, the result will be a scalar field of all directional derivatives in Rn in the directions specified by v; this scalar field will be given in the same coordinate system as is used to interpret expression F. If v is a rooted Vector, the result is the value of the directional derivative of F in the direction of v taken at the root point of v.
If F is a scalar function, the Vector v is normalized. If F is a VectorField, the Vector v is not normalized.
The DirectionalDiff(F,p,dir,c) command computes the directional derivative of F at the point p in the direction dir, where F is interpreted in the coordinate system specified by c, if provided, and otherwise in the current coordinate system. The point p can be a list, a free Vector in Cartesian coordinates or a position Vector. The direction dir can be a free Vector in Cartesian coordinates, a position Vector or a vector field. The result is the value of DirectionalDiff(F,dir,c) evaluated at the point p.
If c is a list of names, the directional derivative of F is taken with respect to these names in the current coordinate system.
If c is an indexed coordinate system, F is interpreted in the combination of that coordinate system and coordinate names.
If c is not specified, F is interpreted in the current coordinate system, whose coordinate name indices define the function's variables.
Note that c has no influence on the interpretation of the direction vector v.
An operator implementing the directional derivative with respect to a VectorField can be obtained using the dot operator with Del, as in V·Del.
Introductory examples where a coordinate system is specified
v1 ≔ 1,2:
W ≔ VectorField⁡u+v,v,cartesianu,v
dd ≔ DirectionalDiff⁡r2,W,polarr,t:
dd ≔ DirectionalDiff⁡VectorField⁡xy,x⁢y,W
Examples where a list of variable names is provided
v2 ≔ 1,0:
dd ≔ DirectionalDiff⁡r⁢cos⁡θ,v2,r,θ:
Examples where the information is given in the form of a Rooted Vector
vs ≔ VectorSpace⁡1,Pi2,polarr,t:
v3 ≔ vs:-Vector⁡1,1
v4 ≔ vs:-Vector⁡0,1
Examples using the dot operator to construct a directional derivative operator
V ≔ VectorField⁡y⁢z,x⁢z,x⁢y
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