# Symbolic Calculations with Units

*Mathematica*'s ability to deal with symbolic expressions, as well as numbers, allows you to use it for many kinds of mathematics. *Mathematica*'s unit system utilizes this symbolic code base, facilitating calculus using Quantity expressions.

Solve[equation,vars] | solution to an equation for x |

Integrate[f,x] | the indefinite integral |

D[f,x] | the (partial) derivative |

Some symbolic mathematical operations that operate on Quantity expressions.

## General

Many symbolic commands are capable of understanding units. Still, there are times when you might wish to add or remove units. These can be handled by *Mathematica*'s general substitution mechanism.

This assigns units to each symbol. Notice that the input symbols are dimensionful, but the output symbols are dimensionless.

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A quick way to test whether an equation is dimensionally consistent is to subtract both sides, substitute in quantity values, and check whether it evaluates to a valid

Quantity object.

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These values are inconsistent with the equation, so messages are issued, and the computation returns only partially evaluated. As a result, the output has

Head Plus.

This substitution replaces each quantity with just the numerical portion, leaving a purely symbolic expression.

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## Solve

Solve is aware of Quantity and will automatically determine the units of unknown variables within the equation. Units for variables can be specified by inserting the variable inside a Quantity expression with the desired units.

This finds the ratio of

per

.

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This gives the two solutions to the equation

. The solutions are given as replacements for

.

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Units can be specified by entering the unknown variable as a

Quantity.

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If

Solve cannot determine the units of an unknown variable, because of ambiguity or a dimensionality conflict, it will return unevaluated.

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## Integrate

Integrate is aware of Quantity and will correctly combine the units of the integrand and integration variables. If the variable of integration is a Quantity object, then its QuantityMagnitude is used as the integration variable, with assumed units equal to its QuantityUnit. In the case of definite integration, units can be inferred from the endpoints, with no need to explicitly declare the units of the integration variable. In general, it is best to enter the input in one of two forms:

The integrand and all the integration variables are

Quantity objects.

Quantity objects only appear in the limits of definite integration.

Input can be entered in other ways, but Integrate will fail if it cannot determine a consistent assignment of units for all expressions.

Calculate displacement over a given time.

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In this form, the variable

is explicitly declared to be measured in seconds.

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Here, the variable

is explicitly declared to be measured in minutes, and the endpoints are automatically converted to the correct units.

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Calculate displacement symbolically.

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Integrate a linear charge density function. Notice that the whole integrand is a

Quantity, allowing an easy combination of the units.

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If the integration variables are not declared as quantities, and quantity objects in the integrand are constants,

Integrate will maintain the separated form of the integrand.

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If you declare a unit for the integration variable, then the result will be returned as a

Quantity object.

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When the integrand has nonconstant

Quantity objects as subexpressions,

Integrate must infer units from the various expressions. If these inferences lead to contradictions,

Integrate will fail.

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## D

D is aware of Quantity and will correctly combine the units of Quantity objects in the expression being differentiated with the units of the differentiation variables.

Compute the derivative of one quantity with respect to another.

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Compute the second derivative of position, the acceleration.

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D will automatically keep track of units.

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Differentiate an expression containing several

Quantity objects.

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When differentiating a sum of

Quantity objects,

Expand can often provide simplification.

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Computing a gradient with respect to

Quantity variables.

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