DirectionalDiff - Maple Help

VectorCalculus

 DirectionalDiff
 computes the directional derivative of a scalar field in the direction given by a vector

 Calling Sequence DirectionalDiff(F,v,c) DirectionalDiff(F,p,dir,c)

Parameters

 F - the scalar or vector field to differentiate v - Vector(algebraic); the direction Vector or vector field p - point=list(algebraic) or point=Vector(algebraic); point where the derivative will be evaluated dir - list(algebraic) or Vector(algebraic); components specifying the direction of the directional derivative in a specified coordinate system c - (optional) list(name) or symbol[name, name, ...]; list of names or name of the coordinate system indexed by the coordinate names

Description

 • 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 ${R}^{n}$ 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 $V·\mathrm{Del}$.

Examples

 > $\mathrm{with}\left(\mathrm{VectorCalculus}\right):$

Introductory examples where a coordinate system is specified

 > $\mathrm{SetCoordinates}\left(\mathrm{cartesian}\left[x,y\right]\right)$
 ${{\mathrm{cartesian}}}_{{x}{,}{y}}$ (1)
 > $\mathrm{v1}≔⟨1,2⟩:$
 > $\mathrm{DirectionalDiff}\left({r}^{2},\mathrm{v1},\mathrm{polar}\left[r,t\right]\right)$
 $\frac{{2}{}{r}{}{\mathrm{cos}}{}\left({t}\right){}\sqrt{{5}}}{{5}}{+}\frac{{4}{}{r}{}{\mathrm{sin}}{}\left({t}\right){}\sqrt{{5}}}{{5}}$ (2)
 > $W≔\mathrm{VectorField}\left(⟨u+v,v⟩,\mathrm{cartesian}\left[u,v\right]\right)$
 ${W}{≔}\left({u}{+}{v}\right){\stackrel{{_}}{{e}}}_{{u}}{+}\left({v}\right){\stackrel{{_}}{{e}}}_{{v}}$ (3)
 > $\mathrm{DirectionalDiff}\left({r}^{2},\mathrm{point}=\left[1,\mathrm{\pi }\right],W,\mathrm{polar}\left[r,t\right]\right)$
 ${2}$ (4)
 > $\mathrm{dd}≔\mathrm{DirectionalDiff}\left({r}^{2},W,\mathrm{polar}\left[r,t\right]\right):$
 > $\mathrm{simplify}\left(\mathrm{eval}\left(\mathrm{dd},\left[r=1,t=\mathrm{\pi }\right]\right)\right)$
 ${2}$ (5)
 > $\mathrm{dd}≔\mathrm{DirectionalDiff}\left(\mathrm{VectorField}\left(⟨\frac{x}{y},xy⟩\right),W\right)$
 ${\mathrm{dd}}{≔}\left(\frac{{x}{+}{y}}{{y}}{-}\frac{{x}}{{y}}\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left(\left({x}{+}{y}\right){}{y}{+}{y}{}{x}\right){\stackrel{{_}}{{e}}}_{{y}}$ (6)

Examples where a list of variable names is provided

 > $\mathrm{DirectionalDiff}\left(pq,⟨1,2⟩,\left[p,q\right]\right)$
 $\frac{{q}{}\sqrt{{5}}}{{5}}{+}\frac{{2}{}{p}{}\sqrt{{5}}}{{5}}$ (7)
 > $\mathrm{v2}≔⟨1,0⟩:$
 > $\mathrm{SetCoordinates}\left(\mathrm{polar}\right)$
 ${\mathrm{polar}}$ (8)
 > $\mathrm{dd}≔\mathrm{DirectionalDiff}\left(r\mathrm{cos}\left(\mathrm{\theta }\right),\mathrm{v2},\left[r,\mathrm{\theta }\right]\right):$
 > $\mathrm{simplify}\left(\mathrm{dd}\right)$
 ${1}$ (9)

Examples where the information is given in the form of a Rooted Vector

 > $\mathrm{SetCoordinates}\left(\mathrm{polar}\left[r,t\right]\right)$
 ${{\mathrm{polar}}}_{{r}{,}{t}}$ (10)
 > $\mathrm{vs}≔\mathrm{VectorSpace}\left(\left[1,\frac{\mathrm{\pi }}{2}\right],\mathrm{polar}\left[r,t\right]\right):$
 > $\mathrm{v3}≔\mathrm{vs}:-\mathrm{Vector}\left(\left[1,1\right]\right)$
 ${\mathrm{v3}}{≔}\left[\begin{array}{c}{1}\\ {1}\end{array}\right]$ (11)
 > $\mathrm{v4}≔\mathrm{vs}:-\mathrm{Vector}\left(\left[0,1\right]\right)$
 ${\mathrm{v4}}{≔}\left[\begin{array}{c}{0}\\ {1}\end{array}\right]$ (12)
 > $\mathrm{DirectionalDiff}\left({r}^{2},\mathrm{v3}\right)$
 $\sqrt{{2}}$ (13)
 > $\mathrm{DirectionalDiff}\left({r}^{2},\mathrm{v4}\right)$
 ${0}$ (14)
 > $\mathrm{SetCoordinates}\left(\mathrm{cartesian}\left[x,y\right]\right)$
 ${{\mathrm{cartesian}}}_{{x}{,}{y}}$ (15)
 > $\mathrm{DirectionalDiff}\left({y}^{2}{x}^{2},\mathrm{point}=\left[1,2\right],⟨0,1⟩,\mathrm{cartesian}\left[x,y\right]\right)$
 ${4}$ (16)
 > $\mathrm{DirectionalDiff}\left({y}^{2}{x}^{2},\mathrm{RootedVector}\left(\mathrm{root}=\left[1,2\right],\left[0,1\right]\right),\mathrm{cartesian}\left[x,y\right]\right)$
 ${4}$ (17)
 > $\mathrm{DirectionalDiff}\left({y}^{2}{x}^{2},\mathrm{RootedVector}\left(\mathrm{root}=\left[1,\frac{\mathrm{\pi }}{2}\right],\left[1,1\right],\mathrm{polar}\left[r,t\right]\right),\mathrm{cartesian}\left[x,y\right]\right)$
 ${0}$ (18)

Examples using the dot operator to construct a directional derivative operator

 > $\mathrm{SetCoordinates}\left(\mathrm{cartesian}\left[x,y,z\right]\right)$
 ${{\mathrm{cartesian}}}_{{x}{,}{y}{,}{z}}$ (19)
 > $V≔\mathrm{VectorField}\left(⟨yz,xz,xy⟩\right)$
 ${V}{≔}\left({y}{}{z}\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left({x}{}{z}\right){\stackrel{{_}}{{e}}}_{{y}}{+}\left({y}{}{x}\right){\stackrel{{_}}{{e}}}_{{z}}$ (20)
 > $\mathrm{normal}\left(\left(V·\mathrm{Del}\right)\left(xyz\right)\right)$
 $\sqrt{{{y}}^{{2}}{}{{x}}^{{2}}{+}{{x}}^{{2}}{}{{z}}^{{2}}{+}{{y}}^{{2}}{}{{z}}^{{2}}}$ (21)
 > $\left(V·\mathrm{Del}\right)\left(\mathrm{VectorField}\left(⟨\frac{1}{x},\frac{1}{y},\frac{1}{z}⟩\right)\right)$
 $\left({-}\frac{{y}{}{z}}{{{x}}^{{2}}}\right){\stackrel{{_}}{{e}}}_{{x}}{+}\left({-}\frac{{x}{}{z}}{{{y}}^{{2}}}\right){\stackrel{{_}}{{e}}}_{{y}}{+}\left({-}\frac{{y}{}{x}}{{{z}}^{{2}}}\right){\stackrel{{_}}{{e}}}_{{z}}$ (22)