Scalar Product Scalar Product of a Vector with itself ? A. A = | A || A | cos 0 º = A 2
Scalar Product Scalar Product of a Vector and Unit vector ? x. A = | x || A | cosα = A x Yields the component of a vector in a direction of the unit vector Where alpha is an angle between A and unit vector x ^^
Spherical Coordinates For many mathematical problems, it is far easier to use spherical coordinates instead of Cartesian ones. In essence, a vector r (we drop the underlining here) with the Cartesian coordinates (x,y,z) is expressed in spherical coordinates by giving its distance from the origin (assumed to be identical for both systems) |r|, and the two angles and between the direction of r and the x- and z-axis of the Cartesian system. This sounds more complicated than it actually is: and are nothing but the geographic longitude and latitude. The picture below illustrates this
Tutorial Evaluate: Where C is right half of the circle : x 2 +y 2 =16 Solution We first need a parameterization of the circle. This is given by, We now need a range of t’s that will give the right half of the circle. The following range of t’s will do this: Now, we need the derivatives of the parametric equations and let’s compute ds:
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