Mathcad 15.0 has had various symbolic calculation capabilities for a long time. These capabilities were absent in Mathcad Prime 1.0 but, we are happy to announce, they have been added back into the available mix of functionality for Mathcad Prime 2.0!
This newly restored symbolic calculation capability in Mathcad Prime 2.0 integrates smoothly with numeric calculation, using patented integration technology. Often in math and engineering software, formulas must be altered before they can be applied to a specific case. Mathcad Prime 2.0’s symbolic calculation functionality automates this process for you and allows you to set up the right form of the formula before plugging in numbers.
Kent Pitman, the Software Development Manager at PTC who oversees work on the calculation engine for Mathcad, shared his views with me on four interesting aspects of symbolic calculation in Mathcad Prime 2.0: explicit evaluation, the unique integration between symbolic and numeric calculations, using Lagrange notation (F’) for derivatives, and Leibniz notation for derivatives.
In Mathcad Prime 2.0, with the explicit evaluation feature, Pitman says, “you can control precisely what variables are looked at within a problem and it’s good for engineers who must show calculations step-by-step. This can be a requirement for certain engineers.” In the following example, he showed me an explicit calculation where he specified which numeric values to use, and which to leave unbound (and effectively handled symbolically).
Next, Pitman showed me how the powerful integration between numerics and symbolics technology can be used. What’s interesting about this feature is that along with being able to use symbolics to define what you’re going to do numerically, you can also accomplish the reverse, and use numerics to inform symbolics. This is a functionality that was standard in Mathcad 15.0 and has been added as one of the advanced features of Mathcad 2.0. In the following example, you can see how symbolics can be used to compute the expression that would become the definition of a function “f”. Then, when the function is called, Mathcad Prime 2.0 automatically computes sin(x), not the integral.
Below is another example of the numerics and symbolics integration. You can see that a symbolic solve is used at definition time and then the result can be used for numeric calculations. You can also see in this example how Mathcad Prime 2.0 draws attention to potential errors, since the value of “a” has no value and the definition of the variable “a” is only “some constant”.
The next feature Pitman and I discussed was the notation for derivatives. This is a newly improved feature that allows you to directly express Lagrange notation for derivatives, rather than having to write a program to implement notation. In contrast to past Mathcad software, which limited where you could use Lagrange notation, you can now use it for any expression where a math expression is called for. Because it can now be a direct expression, it simplifies your calculation. Pitman says, “This is a new feature. We’re not just bringing the software up to date, but we’re adding to it and trying to improve it as well.”
Mathcad has traditionally allowed derivatives to be expressed only using Leibniz notation. The Leibniz notation basically describes a function by talking about its definition. That is, instead of using a notation like the one below, and then referring to “the function F”, one simply referred to “the function sin(x)+x” directly. So, the function that was the derivative of F was referred to by reference to that definition.
Starting with Mathcad Prime 2.0, the Lagrange notation for derivatives is also possible to use. This notation assumes modern functional notation, where functions have names, and uses F’ to denote the derivative of F, as in:
Note that because the function is parameterized, changes of variables are a little easier (see left image) and the function can be easily evaluated at various points (see right image).
Previously one would have written:
Here is another, more complicated example (see left image). In Mathcad Prime 2.0, one can instead write it as it appears in the second image:
Note that both of the above two expressions avoid having the “chain rule” applied to sin(x), as would happen if you used the Leibniz notation this way:
Below you can see how use of explicit evaluation allows a number of useful transformations.
Whether you’re new to Mathcad Prime and see symbolics as a new feature or you’ve got experience with Mathcad 15.0 and see symbolics as a restoration of what belongs there, you’ll have to agree that the presence of symbolics in Mathcad Prime 2.0 is a welcome one. It is considered to be a major tool for many engineers and is a great advanced feature. I’d like to give a special thank you to Kent Pitman for sitting down and sharing all of these exciting highlights with me, and to you for reading. Keep your eyes out for our next post!