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VectorCalculus

 TangentPlane
 compute the plane tangent to a surface

 Calling Sequence TangentPlane(f, var1, var2, var3, outvars, coordopt) TangentPlane(f, pt, outvars, coordopt)

Parameters

 f - free or position Vector, Vector- or scalar-valued procedure, scalar expression or equation; specify the surface var1 - name=algebraic; specify the name and value of the first parameter var2 - name=algebraic; specify the name and value of the second parameter var3 - name=algebraic; (optional) specify the name and value of the third parameter in the case where f is given as equation pt - Vector; in the case where f is given as an equation, the point at which the tangent plane is to be computed can be specified as a 3-element Vector outvars - 'outnames'=list(name); (optional) specify the names of the (3) output (cartesian) coordinate system variables coordopt - 'coordinates'=name[name,name,name]; (optional) specify the coordinate system in which the input f is to be interpreted; this is ignored if f is a Vector with a coordinates or coords attribute; coordinates may be abbreviated as coords

Description

 • The TangentPlane(f, var1, var2) command computes the plane tangent to the surface f at the point specified by var1 and var2. The surface can be entered as a Vector, a Vector-valued procedure, a scalar expression, or a scalar-valued procedure. If f is entered as a procedure, then it must accept at least 2 arguments; it will be evaluated as $f\left(\mathrm{lhs}\left(\mathrm{var1}\right),\mathrm{lhs}\left(\mathrm{var2}\right)\right)$ to obtain the expression for the surface and will be evaluated as $f\left(\mathrm{rhs}\left(\mathrm{var1}\right),\mathrm{rhs}\left(\mathrm{var2}\right)\right)$ to obtain the point of tangency.
 • The TangentPlane(f, var1, var2, var3) command computes the plane tangent to the surface f at the point specified by the three var parameters, where f is defined implicitly by an equation, for example ${x}^{2}+{y}^{2}+{z}^{2}=1$.  No attempt is made to verify that the point specified by the var parameters is actually on the surface.
 • The TangentPlane(f, pt) command computes the plane tangent to the surface f at the point specified by the 3-element Vector pt parameters, where f is defined implicitly by an equation, for example ${x}^{2}+{y}^{2}+{z}^{2}=1$.  No attempt is made to verify that the point specified by the pt parameter is actually on the surface.
 • If f is a scalar expression, an equation, or a scalar-valued procedure, then the current coordinate system must be three-dimensional, e.g., cartesian (in three dimensions), spherical, or cylindrical, or the coordopt option must be provided and specify a three-dimensional coordinate system.
 • If f is a scalar expression, then the second and third arguments must be equations in which the left-hand sides are the coordinate names from the current coordinate system or from the coordopts option. For example, if f is a scalar expression in $x$ and $z$, the ambient coordinate system is ${\mathrm{cartesian}}_{x,y,z}$, and the second and third arguments are $x=a$ and $z=b$, respectively, then f is parameterized as the Vector <$x$, f, $z$>. The tangent plane will be computed at the point $x=a$, $z=\genfrac{}{}{0}{}{f}{}|\genfrac{}{}{0}{}{}{x=a,z=b}$, $z=b$ .
 • If f is a scalar-valued procedure, the second and third arguments must be equations in which the left-hand sides are the coordinate names from the current coordinate system or from the coordopt option. The procedure f will be evaluated at these two names, producing a scalar expression, and the previous paragraph applies.
 • If f is a procedure, the information in var1 will always be substituted into the first parameter of f, and likewise, the information in var2 will always be substituted into the second parameter.
 • If f is a Vector or Vector-valued procedure, var1 and var2 must specify the names of the two parameters of the surface.
 • If f is a Vector or a scalar expression, the output is a position Vector. If f is a procedure, the output is a procedure that evaluates to a position Vector.  If f is described implicitly, as an equation, then the returned tangent plane will be given as an equation.
 • If the input expression f is not a Vector with a coords attribute, it is interpreted in coordinate system specified by the coordopt option.  If this is not given, then the ambient coordinate system (as set, for example, by SetCoordinates) is assumed.
 • If neither the ambient coordinate system nor the coordinate system in which f is interpreted (see previous item) is 3-dimensional cartesian, then the outvars option must be used to provide the names of the output (cartesian) coordinate system variable names.

Examples

 > $\mathrm{with}\left(\mathrm{VectorCalculus}\right):$
 > $\mathrm{SetCoordinates}\left({'\mathrm{cartesian}'}_{x,y,z}\right)$
 ${{\mathrm{cartesian}}}_{{x}{,}{y}{,}{z}}$ (1)
 > $\mathrm{TangentPlane}\left(f\left(x,y\right),x=a,y=b\right)$
 $\left[\begin{array}{c}{x}\\ {y}\\ {{\mathrm{D}}}_{{1}}{}\left({f}\right){}\left({a}{,}{b}\right){}\left({x}{-}{a}\right){+}{{\mathrm{D}}}_{{2}}{}\left({f}\right){}\left({a}{,}{b}\right){}\left({y}{-}{b}\right){+}{f}{}\left({a}{,}{b}\right)\end{array}\right]$ (2)
 > $\mathrm{TangentPlane}\left(⟨s,t,{s}^{2}+{t}^{2}⟩,s=a,t=b\right)$
 $\left[\begin{array}{c}{x}\\ {y}\\ {2}{}{a}{}\left({x}{-}{a}\right){+}{2}{}{b}{}\left({y}{-}{b}\right){+}{{a}}^{{2}}{+}{{b}}^{{2}}\end{array}\right]$ (3)
 > $\mathrm{TangentPlane}\left(\left(s,t\right)→⟨{s}^{2}+{t}^{2},{s}^{2}-{t}^{2},st⟩,s=a,t=b\right)$
 $\left({x}{,}{y}\right){→}{\mathrm{VectorCalculus:-Vector}}{}\left(\left[{x}{,}{y}{,}{-}\frac{{1}}{{8}}{}\frac{{-}\left({2}{}{{a}}^{{2}}{+}{2}{}{{b}}^{{2}}\right){}\left({-}{{a}}^{{2}}{-}{{b}}^{{2}}{+}{x}\right){-}\left({-}{2}{}{{a}}^{{2}}{+}{2}{}{{b}}^{{2}}\right){}\left({-}{{a}}^{{2}}{+}{{b}}^{{2}}{+}{y}\right)}{{a}{}{b}}{+}{a}{}{b}\right]{,}{\mathrm{attributes}}{=}\left[{\mathrm{positionvector}}{,}{\mathrm{coords}}{=}{{\mathrm{cartesian}}}_{{x}{,}{y}{,}{z}}\right]\right)$ (4)
 > $\mathrm{SetCoordinates}\left({'\mathrm{spherical}'}_{r,\mathrm{φ},\mathrm{θ}}\right)$
 ${{\mathrm{spherical}}}_{{r}{,}{\mathrm{φ}}{,}{\mathrm{θ}}}$ (5)
 > $\mathrm{TangentPlane}\left(⟨1,s,t⟩,s=\frac{\mathrm{π}}{4},t=\frac{\mathrm{π}}{2},\mathrm{outnames}=\left[u,v,w\right]\right)$
 $\left[\begin{array}{c}{u}\\ {v}\\ {-}{v}{+}\sqrt{{2}}\end{array}\right]$ (6)
 > $\mathrm{TangentPlane}\left(r=1,⟨1,\frac{\mathrm{π}}{4},\frac{\mathrm{π}}{2}⟩,\mathrm{outnames}=\left[u,v,w\right]\right)$
 $\frac{{1}}{{2}}{}\sqrt{{2}}{}\left({v}{-}\frac{{1}}{{2}}{}\sqrt{{2}}\right){+}\frac{{1}}{{2}}{}\sqrt{{2}}{}\left({w}{-}\frac{{1}}{{2}}{}\sqrt{{2}}\right){=}{0}$ (7)
 > $\mathrm{SetCoordinates}\left({'\mathrm{cartesian}'}_{x,y,z}\right)$
 ${{\mathrm{cartesian}}}_{{x}{,}{y}{,}{z}}$ (8)
 > $\mathrm{TangentPlane}\left(\left(x,y\right)→{x}^{2}+{y}^{2},x=a,y=b\right)$
 $\left({x}{,}{y}\right){→}{\mathrm{VectorCalculus:-Vector}}{}\left(\left[{x}{,}{y}{,}{2}{}{a}{}\left({x}{-}{a}\right){+}{2}{}{b}{}\left({y}{-}{b}\right){+}{{a}}^{{2}}{+}{{b}}^{{2}}\right]{,}{\mathrm{attributes}}{=}\left[{\mathrm{positionvector}}{,}{\mathrm{coords}}{=}{{\mathrm{cartesian}}}_{{x}{,}{y}{,}{z}}\right]\right)$ (9)
 > $\mathrm{TangentPlane}\left({x}^{2}+{y}^{2},x=a,y=b\right)$
 $\left[\begin{array}{c}{x}\\ {y}\\ {2}{}{a}{}\left({x}{-}{a}\right){+}{2}{}{b}{}\left({y}{-}{b}\right){+}{{a}}^{{2}}{+}{{b}}^{{2}}\end{array}\right]$ (10)
 > $\mathrm{~}[\mathrm{normal}]\left(\mathrm{TangentPlane}\left({r}^{2}\mathrm{θ},r=1,\mathrm{θ}=\frac{\mathrm{π}}{4},\mathrm{coords}={\mathrm{spherical}}_{r,\mathrm{φ},\mathrm{θ}}\right)\right)$
 $\left[\begin{array}{c}{x}\\ {y}\\ {-}\frac{{1}}{{2}}{}\frac{\sqrt{{2}}{}\left({\mathrm{π}}{}{x}{+}{\mathrm{π}}{}{y}{-}{2}{}{\mathrm{π}}{-}{6}{}{x}{+}{2}{}{y}\right)}{{2}{+}{\mathrm{π}}}\end{array}\right]$ (11)

Compatibility

 • The VectorCalculus[TangentPlane] command was updated in Maple 17.
 • The var3, pt, outvars and coordopt parameters were introduced in Maple 17.