Symbolic Algebra

Cassiopeia uses MathML content markup to represent equations internally. This means that equations have mathematical meaning and can therefore be processed by the integrated symbolic algebra system. The tight integration with the super-efficient WYSIWYG equation editor gives Cassiopeia a cutting edge.

Taking derivatives

Press Command-y to create a new equation and then type "y", "=", "a", Ctrl-l, "3", "x", Ctrl-h, "3", ... five times cursor right ..."+", "a", Ctrl-l, "2", "x", Ctrl-h, "2", ...
Now Command-Shift-Doubleclick on this equation to open the equation inspector

enter x in the Variable field and click on Differentiate. The following equation is generated automatically.


Repeating this process gives


That's one way of taking derivatives. There is another one. Create a new equation with Command-y and then type "z", "=", Ctrl-d
We have learned in the last chapter that the green background indicates an unsatisfied operator. Cassiopeia can't make sense of what we have typed so far. So insert an "y" in the enumerator cell and "d", "x" in the denominator cell.
This makes sense now, the green background is gone. Press and hold the Ctrl and the Alt key and drag the first equation onto the last one. Cassiopeia creates a copy of the last equation and inserts the dropped one, replacing the y in the enumerator with the right side of the dropped equation.

Now double-click on the last equation to simplify. Do this a couple of times until Cassiopeia reaches the end of its wisdom.
Cassiopeia is pretty talkative when it comes to simplification. In a real document you would probably like to delete some of the intermediary results. But it might be handy in some cases to follow and doublecheck the simplification process closely. Never rely on the wisdom of Cassiopeia alone. Cassiopeia simplifies by applying predefined rules and those rules might not work correctly in all cases. Making this ruleset user extendable in future versions even increases the error-proneness. However, the longer you work with the system the better you know what operations (rules) can be trusted and what should rather be doublechecked by a creative mind.

In the above example we had to explicitly specify a derivation variable. This would not have been necessary if we entered a function instead of an equation. A function requires the specification of a function variable. Create a copy of the function above (just doubleclick on it), then click into the equation and remove the y on the left of the equal sign. Then type "y", "(", "x", ")". The equation should look as follows now.

We have defined a function. Open the equation inspector with Command-Shift-Doubleclick on this formula. It is now no longer necessary to explicitly specify a derivation variable. Just click on Differentiate to take the derivative of the function.
Do this again to get the second derivative.
Being precise - specifying a function variable on the left side - has made the explicit specification of a derviation variable superfluous. Moreover, since Cassiopeia can recognize the functional character of this equation now it can be drawn without further ado. Shift-doubleclick on the equation or alternatively choose SDM - FunctionGrapfh2D from the menu and drag the first function onto the textview of the appearing function graph inspector.


The graph plot is inserted into the document below the doubleclicked function or at the current insertion location. Drag the other two functions y' and y'' from the document into the textview on the 2D graph inspector and change the from value to 0. Press <Enter> after modifying a value to trigger redraw.


Play with the x and y limits until your are satisfied with the graph. Then choose Tools - Colors from the Cassiopeia menu, select colors and drag one color on each of the three functions in the tableview.


You might also want to play with the values of the coefficients. Generate a PDF for your document. You should get something like the following:

Solving Integrals

We want to solve a few integrals now using the integrated symbolic algebra system. Assume you want to calculate the volume of a sphere. Create a new equation with Command-y and type in the infinitesimal volume element in sphere coordinates like so "d", "V", "=", "r", Ctrl-h, "2", Cursor-up, Cursor-right, "s", "i", "n", Ctrl-g "b", "d", "r", ...
Then in a new equation write down the integral like so: "V", "=", Ctrl-i "d", "d", "V"
then Alt-drag the infinitesimal volume element onto this equation. dV is replaced with its assignment.

Doubleclick on this equation. The rules detect multiple differentials dX in the term and therefore transform the equation into the following form.

Now click into the center of this equation and hit cursor up until the inner integral is selected.


Then press Ctrl-y (see Core - Strokes) to upgrade this integral to a determined integral.


Click into the lower limit cell and enter 0. Hit cursor right four times to get into the upper limit cell (or use the mouse). There enter R.

Press cursor up until the middle integral is selected


and hit Ctrl-y to upgrade that as well.


Enter "0" for the lower limit and "2", Ctrl-g p for the upper limit.


Press cursor up until the outer integral is selected. Upgrade that as well and set the limits 0 and pi.


We are done with the creative part. Cassiopeia can handle the rest of the work. Just doubleclick on the equation(s) until the result is presented as shown below.


We have started the above integration with a dintegral (Ctrl-i d). A dintegral has a single cell behind the integral sign and is useful for expressions of the form
This is a rather special case. In most other cases we would probably rather like to start with the normal integral part reachable via Ctrl-i i. Create a new equation, then type "y", "=", Ctrl-i i.
Note that this integral part has two cells, one for the integrand and one for the differential. Fill the body of the integral with "3", "x", Ctrl-h, "3", use the cursor keys to move to the differential cell and type "x".

Doubleclick on this equation. This gets us

Doubleclick again.

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