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of this fact is called Noether s theorem. Let s quickly review the most famous example of a conserved quantity, the conservation of electric charge. The electromagnetic eld tensor F is de ned in terms of the 4-vector potential via F = A A The Maxwell equations with source terms are written as F = J Now it follows from the de nition of F that J = 0 because F = A A F = A A = A A = A A =0 Hence J is a conserved quantity. This is a fact expressed in the famous continuity equation which tells us that + J = 0 t where is the charge density and J is the current density. It follows that charge is conserved. The charge Q is of course de ned using Q = d 3 x Using the continuity equation and taking the surface of integration S to be at in nity we get dQ = d 3x = d 3 x J = J dA = 0 dt t S Hence, charge is conserved. We demonstrated that charge is conserved starting with the equation of motion for the electromagnetic eld. More formally, we can determine what the conserved

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Before we get into the code that will spring into action each time the user presses the button, we need to take care of the second half of the synthesis we began in the previous section. Do you remember how we defined our @property directive with a statement in the header file, and how we left the corresponding @synthesize directive for our implementation file Specifically, we had to have one @property directive for the UILabel and another for the UIImageView, as follows:

FIGURE 3-12

(partial derivatives commute) (relabel dummy indices)

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as rectangles with small tabs at the top. The package name is on the tab or inside the rectangle. The dotted arrows show dependencies. One package depends on another if changes in the other could possibly force changes in the first. Object diagrams show instances instead of classes. They are useful for explaining small pieces with complicated relationships, especially recursive relationships.

quantities are and relate them to symmetries by looking at the action S, or more particularly the lagrangian. This is where Noether s theorem comes into play symmetries in the lagrangian lead to conserved quantities. You can understand Noether s theorem with a simple one-dimensional example. Consider a particle whose motion is described by a lagrangian L (q, q ) where q= dq dt

The sequence diagram shows the explicit series of interactions as they flow through the system to cause the desired objective or result. The sequence view is especially useful in systems with time-dependent functionality (such as real-time applications) and for complex scenarios where time dependencies are critical. It has two dimensions:

@property (nonatomic, retain) IBOutlet UILabel *label; @property (nonatomic, retain) IBOutlet UIImageView *uiImageView;

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The Euler-Lagrange equations are the equations of motion for this system: d L L =0 dt q q Now suppose that the lagrangian is invariant under a symmetry. That is, the form of the lagrangian does not change under a one parameter coordinate transformation t s(t ): q(t ) q( s ) Saying that the lagrangian is invariant under this symmetry means that d L[ q( s ), q( s )] = 0 ds The symmetry of the lagrangian can be written out explicitly using the chain rule as d L dq L dq L[ q( s ), q( s )] = + =0 ds q ds q ds Now let s get to the central idea. Noether s theorem tells us that Q= p is a conserved quantity, that is, dQ =0 dt dq ds

n One that represents time n Another that represents the various objects participating in a sequence of

Usually, only the sequence of events to which the objects of the system are subject is important; in real-time applications, the time axis is an important measurement. This view identifies the roles of the objects in your system through the sequence of states they traverse to accomplish the goal. This view is an eventdriven perspective of the system. The relationships among the roles are not shown. Class and object diagrams present static views. Interaction diagrams are dynamic. They describe how objects collaborate or interact. A sequence diagram is an interaction diagram that details the functionality and messages (requests and responses) and their timing. The time progresses as you move down the page. The objects involved in the operation are listed from left to right according to when they take part in the message sequence. Figure 3-13 shows a sequence diagram that illustrates the software calls and hardware used to service the calls in a sequence of time, with synchronous messages between each object in the diagram. Each vertical dotted line in Figure 3-13 is a lifeline, representing the time that an object exists. Each arrow is a message call. An arrow goes from the sender to the top of the activation bar of the message on the receiver's lifeline. The activation bar represents the duration of execution of the message. The sequence diagram can have a clarifying note, text inside a dog-eared rectangle. Notes can be put into any kind of UML diagram. The UML uses the following message conventions for sequence diagrams:

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