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Handout_1 - EE 305 Review Prof Shanker Balasubramaniam 1...

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EE 305 Review Prof. Shanker Balasubramaniam 1 Basic Vector Calculus and Algebra 1.1 Notations Cartesian Coordinates ˆ x , ˆ y , ˆ z Cylindrical Coordinates ˆ ρ , ˆ φ , ˆ z Spherical Coordinates ˆ r , ˆ θ , ˆ φ Position Vector r = x ˆ x + y ˆ y + z ˆ z NOTE: All vector will be denoted using either boldface or a bar, eg., a or ¯ a . All dyads (a term that shall be explained shortly) will be identified using upper case letters as in A or using a bar over an uppercase letter ¯ A . There are two kinds of notation that is commonly used. The direct form that was mentioned earlier, and the index notation. The index notation is sometimes easier to understand and manipulate. For instance, a vector can be denoted either as a or equivalently a i for i = 1 , 2 , 3 (we are assuming three dimensional space). One way of interpreting this is to assume that i = 1 = ˆ x , i = 2 = ˆ y and i = 3 = ˆ z , i.e., each number stands for a component. While the index notation is mathematically useful, it should be remembered that each index corresponds to a mapping of a vector on to a directional basis that are used, and hence, are coordinate dependent. The direct form is coordinate independent and represents the quantity as is.
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