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Notes+2+ Fall 2004

# Notes+2+ Fall 2004 - Notes 2 Fall 2004 Scalar Fields and...

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Notes 2 Fall 2004.doc Scalar Fields and Vector Fields A voltage as a function of time may be written as ( ) t v . This is an example of a function of one variable. In electromagnetics this, as well as other quantities of interest, are also often functions of time and space. In the Cartesian coordinate system we might write ( ) t , z , y , x v v The fact that this quantity is now a function of three dimensional space makes this a field . The fact that this quantity is a scalar (there is no direction associated with it) as opposed to a vector, makes this a scalar field . Another good example of a scalar field is the temperature at different locations in the atmosphere. A hot air balloonist, for instance, traveling between Baton Rouge and New Orleans might be concerned about the temperatures at different altitudes ( ) z , at different points, along his route. ( y x , ) Since vector quantities are often of interest (force, velocity, momentum, current density, electric and magnetic fields …) it is reasonable to want to consider vector fields, ( ) t , z , y , x E E G G G Choosing the symbol E for this example is appropriate because we are thinking about the electric field which is a vector quantity with both a magnitude and direction. Staying with the Cartesian coordinate system we can decompose this vector field into its components like this, () ( ) ( )( z ˆ t , z , y , x y ˆ t , z , y , x x ˆ t , z , y , x t , z , y , x z y x E E E E ) + + = G The three components of , , , are themselves, scalar fields, and so we can say that to describe the information in a vector field we need to keep track of three scalar fields. , , and are unit vectors (of length one) and balance the vector nature of the left hand side of the equation with the right hand side. E G x E y E z E x ˆ y ˆ z ˆ Taking derivatives of fields: The del operator, the gradient, the divergence and the curl. Differentiating scalar and vector fields: () [] ( ) t t , z , y , x t , z , y , x dt d = v v () [] ( ) ( ) ( ) z ˆ t t , z , y , x y ˆ t t , z , y , x x ˆ t t , z , y , x t , z , y , x dt d z y x + + = E E E E G 1

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Notes 2 Fall 2004.doc We can also take the derivative with respect to a particular direction in space: For the scalar field, () [] ( ) x t , z , y , x t , z , y , x dx d = v v For the vector field, () [] () ( ) ( ) z ˆ x t , z , y , x y ˆ x t , z , y , x x ˆ x t , z , y , x t , z , y , x dx d z y x + + = E E E E G see example 1 below Remember that a derivative gives us information about how a function is changing with respect to some variable (usually a direction or time). For a scalar field, as shown in the example above we can take the derivative with respect to one direction, or we can create an expression that shows how the scalar field is changing in each of the three directions by adding together the derivatives in each direction along with the unit vector for each term (which results in a vector
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Notes+2+ Fall 2004 - Notes 2 Fall 2004 Scalar Fields and...

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