Smart Popups and Drag to Solve - Maple Help

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 Smart Pop-ups and Drag-to-Solve

The Smart Pop-ups facility is a convenient Clickable Math mechanism for suggesting common operations, algebraic manipulations, and graphs of 2D Math expressions. The menu items are particularly suitable for students, and can be used to demonstrate mathematical identities without having to manually type in commands.

These menus are available by selecting either an entire expression or just a portion of an expression with the mouse pointer. The pop-up menu is generated and displayed automatically, and disappear when you click anywhere off of the pop-up menu.

The Smart Pop-ups menus have been augmented in Maple 17 with new choices such as completion of the square and simplification based on expression size. The suggestions offered in the pop-up menu are now computed with time limits to avoid lengthy delays.

Various other aspects of the mechanism which generates and labels the suggestions have been fine-tuned. A displayed suggested result of applying the $\mathrm{simplify}$ command will now visually match the result of applying the current binding of $\mathrm{simplify}$. For example, if the $\mathrm{RealDomain}$ package has been loaded then a suggested result due to applying simplify will now match the result that $\mathrm{RealDomain}:-\mathrm{simplify}$ would produce.

Complete the square

 > $\mathrm{expand}\left({\left(A+B+C\right)}^{2}\right)$
 ${{A}}^{{2}}{+}{2}{}{A}{}{B}{+}{2}{}{A}{}{C}{+}{{B}}^{{2}}{+}{2}{}{B}{}{C}{+}{{C}}^{{2}}$ (1.1)

Selecting the output expression above produces the Smart Pop-up menu after a brief pause. Completing the square on the given example may be done in terms of one or several of the variables. If the mouse pointer hovers over the appropriate item then the relevant submenu appears. For individual action items (choices within that submenu) a tooltip will further describe the suggested operation.

Below is a screenshot of the Smart Pop-up menu produced for the previous output.

Simplify,size

For some expressions the concept of simplification may depend on context or preference. Simplification in a mathematical sense may not naturally be the same as simplification according to expression length. The menu suggestions will offer the results from the simplify command both with and without its $\mathrm{size}$ option.

Note that duplicate results should be removed. If multiple mechanisms would normally compute the same result then the menu suggestions attempt to show the action which would usually be more efficient. For example, a suggestion to apply the normal command would take precedence over a suggestion to apply the simplify command.

For the following example, the result of applying simplify differs from that of applying simplify,size.

In a Standard Maple Document, selecting a suggested item from the Smart Pop-up menu inserts an arrow followed by the new result. In a Standard Maple Worksheet, the command which produces the result, and the result itself, will be both inserted into a new Execution Group following the current cursor position.

 > $\frac{1{ⅇ}^{-\frac{1{x}^{2}}{4}}{2}^{\frac{1}{4}}{x}^{\frac{3}{2}}}{4}+\frac{1{ⅇ}^{\frac{1{x}^{2}}{4}}{2}^{\frac{3}{4}}\sqrt{x}\sqrt{\mathrm{π}}\mathrm{erf}\left(\frac{1\sqrt{2}x}{2}\right)}{8}+\frac{1{ⅇ}^{\frac{1{x}^{2}}{4}}{2}^{\frac{3}{4}}{x}^{\frac{5}{2}}\sqrt{\mathrm{π}}\mathrm{erf}\left(\frac{1\sqrt{2}x}{2}\right)}{8}$
 $\frac{{1}}{{4}}{}{{ⅇ}}^{{-}\frac{{1}}{{4}}{}{{x}}^{{2}}}{}{{2}}^{{1}{/}{4}}{}{{x}}^{{3}{/}{2}}{+}\frac{{1}}{{8}}{}{{ⅇ}}^{\frac{{1}}{{4}}{}{{x}}^{{2}}}{}{{2}}^{{3}{/}{4}}{}\sqrt{{x}}{}\sqrt{{\mathrm{π}}}{}{\mathrm{erf}}{}\left(\frac{{1}}{{2}}{}\sqrt{{2}}{}{x}\right){+}\frac{{1}}{{8}}{}{{ⅇ}}^{\frac{{1}}{{4}}{}{{x}}^{{2}}}{}{{2}}^{{3}{/}{4}}{}{{x}}^{{5}{/}{2}}{}\sqrt{{\mathrm{π}}}{}{\mathrm{erf}}{}\left(\frac{{1}}{{2}}{}\sqrt{{2}}{}{x}\right)$ (2.1)
 > $\text{# simplify, size 1/4*exp(-1/4*x^2)*2^(1/4)*x^(3/2)+1/8*exp(1/4*x^2)*2^(3/4)*sqrt(x)*sqrt(Pi)*erf(1/2*sqrt(2)*x)+1/8*exp(1/4*x^2)*2^(3/4)*x^(5/2)*sqrt(Pi)*erf(1/2*sqrt(2)*x) SubexpressionMenu:-simplify(1/4*exp(-1/4*x^2)*2^(1/4)*x^(3/2)+1/8*exp(1/4*x^2)*2^(3/4)*sqrt(x)*sqrt(Pi)*erf(1/2*sqrt(2)*x)+1/8*exp(1/4*x^2)*2^(3/4)*x^(5/2)*sqrt(Pi)*erf(1/2*sqrt(2)*x),size);}$
 $\frac{{1}}{{8}}{}{{2}}^{{1}{/}{4}}{}\sqrt{{x}}{}\left(\sqrt{{2}}{}\sqrt{{\mathrm{π}}}{}{\mathrm{erf}}{}\left(\frac{{1}}{{2}}{}\sqrt{{2}}{}{x}\right){}\left({1}{+}{{x}}^{{2}}\right){}{{ⅇ}}^{\frac{{1}}{{4}}{}{{x}}^{{2}}}{+}{2}{}{{ⅇ}}^{{-}\frac{{1}}{{4}}{}{{x}}^{{2}}}{}{x}\right)$ (2.2)

Trigonometric identities

The Smart Pop-up menus provide an easy way for students to prove trigonometric identities in a self-documenting step-by-step manner.

The following operations are all achieved by suggestions in the Smart Pop-up menus. This example was executed in a Standard Maple Document, using default 2-D Math input and the self-documenting feature of context-sensitive menu actions.

$\mathrm{sin}\left(3a\right)=3\mathrm{sin}\left(a\right)-4{\mathrm{sin}\left(a\right)}^{3}$

 ${\mathrm{sin}}{}\left({3}{}{a}\right){=}{3}{}{\mathrm{sin}}{}\left({a}\right){-}{4}{}{{\mathrm{sin}}{}\left({a}\right)}^{{3}}$ (3.1)

$\stackrel{\text{angle reduction identity}}{\to }$

 ${2}{}{\mathrm{cos}}{}\left({a}\right){}{\mathrm{sin}}{}\left({2}{}{a}\right){-}{\mathrm{sin}}{}\left({a}\right){=}{3}{}{\mathrm{sin}}{}\left({a}\right){-}{4}{}{{\mathrm{sin}}{}\left({a}\right)}^{{3}}$ (3.2)

$\stackrel{\text{double angle identity: sin(2*a)=2*sin(a)*cos(a)}}{\to }$

 ${4}{}{{\mathrm{cos}}{}\left({a}\right)}^{{2}}{}{\mathrm{sin}}{}\left({a}\right){-}{\mathrm{sin}}{}\left({a}\right){=}{3}{}{\mathrm{sin}}{}\left({a}\right){-}{4}{}{{\mathrm{sin}}{}\left({a}\right)}^{{3}}$ (3.3)

$\stackrel{\text{add sin(a) to both sides}}{\to }$

 ${4}{}{{\mathrm{cos}}{}\left({a}\right)}^{{2}}{}{\mathrm{sin}}{}\left({a}\right){=}{4}{}{\mathrm{sin}}{}\left({a}\right){-}{4}{}{{\mathrm{sin}}{}\left({a}\right)}^{{3}}$ (3.4)

$\stackrel{\text{divide both sides by sin(a)}}{\to }$

 ${4}{}{{\mathrm{cos}}{}\left({a}\right)}^{{2}}{=}\frac{{4}{}{\mathrm{sin}}{}\left({a}\right){-}{4}{}{{\mathrm{sin}}{}\left({a}\right)}^{{3}}}{{\mathrm{sin}}{}\left({a}\right)}$ (3.5)

$\stackrel{\text{normal 1/sin(a)*(4*sin(a)-4*sin(a)^3)}}{\to }$

 ${4}{}{{\mathrm{cos}}{}\left({a}\right)}^{{2}}{=}{4}{-}{4}{}{{\mathrm{sin}}{}\left({a}\right)}^{{2}}$ (3.6)

$\stackrel{\text{divide both sides by 4}}{\to }$

 ${{\mathrm{cos}}{}\left({a}\right)}^{{2}}{=}{1}{-}{{\mathrm{sin}}{}\left({a}\right)}^{{2}}$ (3.7)

$\stackrel{\text{Pythagoras identity: cos(a)^2=1-sin(a)^2}}{\to }$

 ${1}{-}{{\mathrm{sin}}{}\left({a}\right)}^{{2}}{=}{1}{-}{{\mathrm{sin}}{}\left({a}\right)}^{{2}}$ (3.8)