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Minimizing SP (by minimizing the area of the worldsheet) gives us the equations of motion for the X ( , ), and hence the dynamics of the string. In the quiz at the end of Chap. 2 in Prob. 4, you were invited to show that the Polyakov and NambuGoto actions were equivalent by considering the energy-momentum or stress-energy tensor T which is given by T = 2 T 1 SP h h (3.3)

Use case diagrams are closely connected to scenarios. A scenario is an example of what happens when someone interacts with the system. For example, here is a scenario for a security trade: a trader accesses an Internet-based system and chooses the type of security he or she wants to place a trade order to buy or sell (see Figure 3-10).

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In this book we ll go mostly by the name energy-momentum tensor. In a nutshell, the energy-momentum tensor describes the density and ux of energy and momentum in space-time. You should be familiar with the basics of what T is from some exposure to or study of quantum eld theory, so we re just going to go with that and describe how it works in string theory. When working out the solution to Prob. 4 in Chap. 2, you should have found that 1 T = X X h h X X 2

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FIGURE 3-10

Figure 3-10 shows a trade use case for the online trading site. The actor is a trader. The connection between actor and use case is a communication association. Actors are represented by stick figures. Use cases are represented by ovals. A common issue regarding drawing use cases is having two actions tied to each other, essentially showing a flowchart. In Figure 3-10, the case study trading system menu is invoked for the Order entry and browse functionality and subsequent calls to the Stratus CTPS and Oracle database. Lines that link actors to use cases represent communications. A use case diagram is a collection of actors, use cases, and their communications. A single use case can have multiple actors. A system boundary rectangle separates the system from the external actors. A use case generalization shows that one use case is a special kind of another use case. Use case diagrams are important to use when you are

(3.4)

n Determining new requirements n Communicating with clients their simplicity makes use case diagrams a

The result should look like this:

The rst property that we will establish for the energy-momentum tensor is that it has zero trace. We can calculate the trace using the induced metric: Tr (T ) = T = h T

A class diagram provides an overview of a system by showing its classes and the relationships among them. Class diagrams are static; they display what interacts but not what happens when they do interact. The class diagram shown in Figure 3-11 models an EJB session bean used to order equities from a securities market. The central method is makeOrder, which creates and returns an OrderBean. Associated with it is the makeUser, which creates and returns a UserBean. UML class notation is a rectangle divided into three parts: class name, attributes, and operations. Names of abstract classes, such as com.ucny.trading.ejb.sessionbeans.AccessData, are in italics. Relationships among classes are the connecting links. A class diagram can have three kinds of relationships:

The trace is easy to calculate:

association exists between two classes if an instance of one class must know about the other to perform its work. In a diagram, an association is a link connecting two classes.

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= h X X h X X = h X X h X X =0

FIGURE 3-11

another. A generalization shows a triangle pointing to the superclass. An association has two ends. An end may include a role name to clarify the nature of the association. For example, an OrderDetail is a line item of each Order.

(because h h = = 2)

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