 ParametrizeCurve - Maple Help

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Physics[Vectors][ParametrizeCurve] - parametrize, in terms of a single parameter, curves given in vectorial or algebraic form, or passed as a set of algebraic equations

Physics[Vectors][ParametrizeSurface] - parametrize, in terms of two parameters, surfaces given in vectorial or algebraic form, or passed as a set of algebraic equations

Physics[Vectors][ParametrizeVolume] - parametrize, in terms of three parameters, volumes given in vectorial or algebraic form, or passed as a set of algebraic equations Calling Sequence ParametrizeCurve(R) ParametrizeCurve(R, basis = ..., parameters = ..., output = ...) ParametrizeSurface(R) ParametrizeSurface(R, basis = ..., parameters = ..., output = ...) ParametrizeVolume(R) ParametrizeVolume(R, basis = ..., parameters = ..., output = ...) Parameters

 R - a region of 3D space, represented either in vectorial form, or as an equation or algebraic expression (understood to be equal to 0) relating the coordinates, or as a set or list of them basis = ... - optional, default value is derived from R; the right-hand side is one of the number identifiers {1, 2, 3}, or one of the related keywords cartesian, cylindrical, or spherical, so that the returned parametric equations are expressed solved for the coordinates of the indicated basis, or when output = vector the vectorial parametric equation is written in that basis. parameters = ... - synonym parameter; optional, default value is derived from R in different ways; the right-hand side can be a set or list with 1, 2 or 3 parameters of type name output = ... - optional, default value is standard, the parametrization is expressed as an ordered list of equations where the left-hand side are the coordinates (cartesian, cylindrical or spherical) as functions of the parameters, and the right-hand sides are algebraic expression involving only the parameters. An alternative value is vector, so that the parametric equations are transformed into a vector function and returned as such Description

 • ParametrizeCurve, ParametrizeSurface and ParametrizeVolume receive R representing a region of 3D space, either in vectorial form (see notation), or as an algebraic expression, or as a set or list of algebraic equations relating the coordinates, and return a sequence of two objects C, T, where C is an ordered list of equations solved for one of the coordinate systems, cartesian, cylindrical or spherical, and T is a list with the parameters used, or the single parameter itself in the case of ParametrizeCurve.
 • The returned list of equations C has the coordinates on the left-hand sides expressed as functions of the parameters and the right-hand sides as algebraic expressions involving only the parameters. The ordering of the parameters in the functions on the left-hand sides of C is the same ordering of the parameters in T. NOTE: ParametrizeSurface and ParametrizeVolume are used by int to compute surface and volume integrals, in which case the ordering of the parameters in T is relevant, since the surface and volume elements of the integral involve a cross product of vectors.
 • You can optionally request the C in the output to be expressed in vectorial form instead of a list of parametric equations. For that purpose, pass the optional argument output = vector. This vectorial form is constructed using the right-hand sides of the parametric equations as coefficients multiplied by the unit vectors of the corresponding vector basis.
 • By default, the vector basis (cartesian, cylindrical or spherical), related system of coordinates and parameters used to express the output are derived from the given region R. You can optionally change all that using the corresponding optional arguments, basis and parameters.
 • The basis option induces a call to ChangeBasis on R. So if R is passed in vectorial form it is first rewritten in the indicated basis. Also, if output = vector, then the vector is expressed in the basis indicated, and if output = standard (default value), the list of equations returned have on the left-hand sides the coordinates of the indicated basis.
 • The parameters = ... option, where the right-hand side is a set or a list of names to be used to parametrize the region R, should include only one, two or three parameters, respectively for ParametrizeCurve, ParametrizeSurface and ParametrizeVolume. When the parameters are indicated as a set they are automatically ordered as a list with the canonical ordering for coordinate systems, which is $x,y,z$ for cartesian, $\mathrm{\rho },\mathrm{\phi },z$ for cylindrical and $r,\mathrm{\theta },\mathrm{\phi }$ for spherical. It is possible to use coordinates of one of these three coordinate systems as parameters, in which case the geometrical relation between coordinates of the different systems is automatically taken into account. This is practical, for example, when parametrizing expressions that include ${x}^{2}+{y}^{2}$, so indicating the use of $\mathrm{\phi }$ as parameter automatically induces the use of $x=\mathrm{\rho }\mathrm{cos}\left(\mathrm{\phi }\right),y=\mathrm{\rho }\mathrm{sin}\left(\mathrm{\phi }\right)$.
 • These three parametrization commands are used by int to perform vector path, surface and volume integrals. Examples

 > $\mathrm{with}\left({\mathrm{Physics}}_{\mathrm{Vectors}}\right)$
 $\left[{\mathrm{&x}}{,}{\mathrm{+}}{,}{\mathrm{.}}{,}{\mathrm{Assume}}{,}{\mathrm{ChangeBasis}}{,}{\mathrm{ChangeCoordinates}}{,}{\mathrm{CompactDisplay}}{,}{\mathrm{Component}}{,}{\mathrm{Curl}}{,}{\mathrm{DirectionalDiff}}{,}{\mathrm{Divergence}}{,}{\mathrm{Gradient}}{,}{\mathrm{Identify}}{,}{\mathrm{Laplacian}}{,}{\nabla }{,}{\mathrm{Norm}}{,}{\mathrm{ParametrizeCurve}}{,}{\mathrm{ParametrizeSurface}}{,}{\mathrm{ParametrizeVolume}}{,}{\mathrm{Setup}}{,}{\mathrm{Simplify}}{,}{\mathrm{^}}{,}{\mathrm{diff}}{,}{\mathrm{int}}\right]$ (1)

Consider the following $C$ representing a curve in space

 > $C≔\left\{y={x}^{2},z=0\right\}$
 ${C}{≔}\left\{{y}{=}{{x}}^{{2}}{,}{z}{=}{0}\right\}$ (2)

The parametric equations for this curve are

 > $\mathrm{ParametrizeCurve}\left(C\right)$
 $\left[{x}{}\left({t}\right){=}{t}{,}{y}{}\left({t}\right){=}{{t}}^{{2}}{,}{z}{}\left({t}\right){=}{0}\right]{,}{t}$ (3)

The right-hand sides of the equations above are the components of the position vector $\stackrel{\to }{r}$ in cartesian coordinates, from where a vectorial form of these equations is

 > $\mathrm{ParametrizeCurve}\left(C,\mathrm{output}=\mathrm{vector}\right)$
 ${{t}}^{{2}}{}\stackrel{{\wedge }}{{j}}{+}{t}{}\stackrel{{\wedge }}{{i}}{,}{t}$ (4)

The curve $C$ can also be passed in vector form

 > $C≔x\mathrm{_i}+{x}^{2}\mathrm{_j}$
 ${C}{≔}{{x}}^{{2}}{}\stackrel{{\wedge }}{{j}}{+}{x}{}\stackrel{{\wedge }}{{i}}$ (5)
 > $\mathrm{ParametrizeCurve}\left(C\right)$
 $\left[{x}{}\left({t}\right){=}{t}{,}{y}{}\left({t}\right){=}{{t}}^{{2}}{,}{z}{}\left({t}\right){=}{0}\right]{,}{t}$ (6)

The equations of circle of radius $a$ on the ($x,y$) plane can be written as

 > $C≔\left\{{x}^{2}+{y}^{2}={a}^{2},z=0\right\}$
 ${C}{≔}\left\{{z}{=}{0}{,}{{x}}^{{2}}{+}{{y}}^{{2}}{=}{{a}}^{{2}}\right\}$ (7)
 > $\mathrm{ParametrizeCurve}\left(C\right)$
 $\left[{x}{}\left({\mathrm{\phi }}\right){=}{a}{}{\mathrm{cos}}{}\left({\mathrm{\phi }}\right){,}{y}{}\left({\mathrm{\phi }}\right){=}{a}{}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){,}{z}{}\left({\mathrm{\phi }}\right){=}{0}\right]{,}{\mathrm{\phi }}$ (8)

In vector notation,

 > $\mathrm{ParametrizeCurve}\left(C,\mathrm{output}=\mathrm{vector}\right)$
 ${a}{}{\mathrm{cos}}{}\left({\mathrm{\phi }}\right){}\stackrel{{\wedge }}{{i}}{+}{a}{}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){}\stackrel{{\wedge }}{{j}}{,}{\mathrm{\phi }}$ (9)

Changing the basis of this vector, for example to cylindrical, we get

 > $\mathrm{ChangeBasis}\left({1}_{},\mathrm{cylindrical}\right)$
 ${a}{}\stackrel{{\wedge }}{{\mathrm{\rho }}}$ (10)

The same result can be obtained by specifying the basis

 > $\mathrm{ParametrizeCurve}\left(C,\mathrm{output}=\mathrm{vector},\mathrm{basis}=\mathrm{cylindrical}\right)$
 ${a}{}\stackrel{{\wedge }}{{\mathrm{\rho }}}{,}{\mathrm{\phi }}$ (11)

An alternative parametrization of $C$ can be computed by indicating a parameter different from the natural polar coordinates $\mathrm{\phi }$, for example (any letter serves the purpose)

 > $\mathrm{ParametrizeCurve}\left(C,\mathrm{parameter}=\mathrm{τ}\right)$
 $\left[{x}{}\left({\mathrm{\tau }}\right){=}{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{-}{{a}}^{{2}}{+}{{\mathrm{\tau }}}^{{2}}\right){,}{y}{}\left({\mathrm{\tau }}\right){=}{\mathrm{\tau }}{,}{z}{}\left({\mathrm{\tau }}\right){=}{0}\right]{,}{\mathrm{\tau }}$ (12)

In general, the parametrization is attempted in the coordinates received. For example, rewriting $C=\left\{z=0,{x}^{2}+{y}^{2}={a}^{2}\right\}$ in cylindrical coordinates

 > $\mathrm{ChangeCoordinates}\left(C,\mathrm{cylindrical}\right)$
 $\left\{{z}{=}{0}{,}{{\mathrm{\rho }}}^{{2}}{=}{{a}}^{{2}}\right\}$ (13)
 > $\mathrm{ParametrizeCurve}\left(\right)$
 $\left[{\mathrm{\rho }}{}\left({t}\right){=}{-}{a}{,}{\mathrm{\phi }}{}\left({t}\right){=}{t}{,}{z}{}\left({t}\right){=}{0}\right]{,}{t}$ (14)

The special case of a straight line that passes through two given points $\stackrel{\to }{A}$ and $\stackrel{\to }{B}$. Consider

 > $A≔2\mathrm{_i}-3\mathrm{_j}+4\mathrm{_k}$
 ${A}{≔}{2}{}\stackrel{{\wedge }}{{i}}{-}{3}{}\stackrel{{\wedge }}{{j}}{+}{4}{}\stackrel{{\wedge }}{{k}}$ (15)
 > $B≔-4\mathrm{_i}+2\mathrm{_j}-\mathrm{_k}$
 ${B}{≔}{-}{4}{}\stackrel{{\wedge }}{{i}}{+}{2}{}\stackrel{{\wedge }}{{j}}{-}\stackrel{{\wedge }}{{k}}$ (16)
 > $\mathrm{ParametrizeCurve}\left(\mathrm{line}\left(A,B\right)\right)$
 $\left[{x}{}\left({t}\right){=}{2}{-}{6}{}{t}{,}{y}{}\left({t}\right){=}{-}{3}{+}{5}{}{t}{,}{z}{}\left({t}\right){=}{-}{5}{}{t}{+}{4}\right]{,}{t}$ (17)

The values of $t$ that recover $\stackrel{\to }{A}$ and $\stackrel{\to }{B}$ from these parametric equations are, respectively, $t=4$ and $t=-1$.

ParametrizeCurve is used by int when computing path integrals. For example, integrate the number 1 along the line that joints $\stackrel{\to }{A}$ and $\stackrel{\to }{B}$ from ${P}_{1}$ to ${P}_{2}$ (these two points also belonging to that line)

 > $\mathit{P__1}≔\frac{16\mathrm{_i}}{5}-4\mathrm{_j}+5\mathrm{_k}$
 $\mathrm{P__1}{≔}\frac{{16}{}\stackrel{{\wedge }}{{i}}}{{5}}{-}{4}{}\stackrel{{\wedge }}{{j}}{+}{5}{}\stackrel{{\wedge }}{{k}}$ (18)
 > $\mathit{P__2}≔-\frac{44\mathrm{_i}}{5}+6\mathrm{_j}-5\mathrm{_k}$
 $\mathrm{P__2}{≔}{-}\frac{{44}{}\stackrel{{\wedge }}{{i}}}{{5}}{+}{6}{}\stackrel{{\wedge }}{{j}}{-}{5}{}\stackrel{{\wedge }}{{k}}$ (19)
 > $\left(\mathrm{Int}=\mathrm{int}\right)\left(1,\mathrm{r_}=\mathit{P__1}..\mathit{P__2},\mathrm{path}=\mathrm{line}\left(A,B\right)\right)$
 ${\mathrm{Int}}{}\left({1}{,}{\mathrm{r_}}{=}\frac{{16}}{{5}}{}{\mathrm{_i}}{-}{4}{}{\mathrm{_j}}{+}{5}{}{\mathrm{_k}}{..}{-}\frac{{44}}{{5}}{}{\mathrm{_i}}{+}{6}{}{\mathrm{_j}}{-}{5}{}{\mathrm{_k}}{,}{\mathrm{path}}{=}{\mathrm{line}}{}\left({2}{}{\mathrm{_i}}{-}{3}{}{\mathrm{_j}}{+}{4}{}{\mathrm{_k}}{,}{-}{4}{}{\mathrm{_i}}{+}{2}{}{\mathrm{_j}}{-}{\mathrm{_k}}\right)\right){=}{-}{12}{}{\mathrm{_i}}{+}{10}{}{\mathrm{_j}}{-}{10}{}{\mathrm{_k}}$ (20)

ParametrizeSurface and ParametrizeVolume work in the same way as ParametrizeCurve, only with two and three parameters respectively.

 > $\mathit{C__2}≔{x}^{2}+{y}^{2}+{z}^{2}={a}^{2}$
 $\mathrm{C__2}{≔}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}{=}{{a}}^{{2}}$ (21)
 > $\mathrm{ParametrizeSurface}\left(\mathit{C__2}\right)$
 $\left[{x}{}\left({\mathrm{\theta }}{,}{\mathrm{\phi }}\right){=}{a}{}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}{\mathrm{cos}}{}\left({\mathrm{\phi }}\right){,}{y}{}\left({\mathrm{\theta }}{,}{\mathrm{\phi }}\right){=}{a}{}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){,}{z}{}\left({\mathrm{\theta }}{,}{\mathrm{\phi }}\right){=}{a}{}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)\right]{,}\left[{\mathrm{\theta }}{,}{\mathrm{\phi }}\right]$ (22)
 > $\mathit{C__3}≔{x}^{2}+{y}^{2}+{z}^{2}={r}^{2}$
 $\mathrm{C__3}{≔}{{x}}^{{2}}{+}{{y}}^{{2}}{+}{{z}}^{{2}}{=}{{r}}^{{2}}$ (23)
 > $\mathrm{ParametrizeVolume}\left(\mathit{C__3}\right)$
 $\left[{x}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){=}{r}{}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}{\mathrm{cos}}{}\left({\mathrm{\phi }}\right){,}{y}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){=}{r}{}{\mathrm{sin}}{}\left({\mathrm{\theta }}\right){}{\mathrm{sin}}{}\left({\mathrm{\phi }}\right){,}{z}{}\left({r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right){=}{r}{}{\mathrm{cos}}{}\left({\mathrm{\theta }}\right)\right]{,}\left[{r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right]$ (24)
 > $\mathrm{ParametrizeVolume}\left(\mathit{C__3},\mathrm{basis}=\mathrm{spherical},\mathrm{output}=\mathrm{vector}\right)$
 ${r}{}\stackrel{{\wedge }}{{r}}{,}\left[{r}{,}{\mathrm{\theta }}{,}{\mathrm{\phi }}\right]$ (25)
 > $\mathrm{ParametrizeVolume}\left(\mathit{C__3},\mathrm{basis}=\mathrm{cylindrical},\mathrm{output}=\mathrm{vector}\right)$
 $\stackrel{{\wedge }}{{k}}{}{z}{+}{\mathrm{\rho }}{}\stackrel{{\wedge }}{{\mathrm{\rho }}}{,}\left[{\mathrm{\rho }}{,}{\mathrm{\phi }}{,}{z}\right]$ (26) See Also Compatibility

 • The Physics[Vectors][ParametrizeCurve], Physics[Vectors][ParametrizeSurface] and Physics[Vectors][ParametrizeVolume] commands were introduced in Maple 2022.