Chapter01-page7 - will be another vector. The resulting...

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Physics 235 Chapter 1 - 7 - A B × C ( ) = B C × A ( ) = C A × B ( ) Figure 3. Properties of the vector product between the vectors A and B . Differentiation and Integration Two important operations on both scalars and vectors are differentiation and integration. These operations are used to define important mechanical quantities (such as velocity and acceleration), and a thorough understanding of operations involving differentiation and integration is required in order to succeed in this course. Scalar Differentiation . We can differentiate vectors and scalars with respect to a scalar variable s . o The result of the differentiation of a scalar with respect to another scalar variable will be another scalar. The result of the differentiation will be independent of the coordinate system. o The result of the differentiation of a vector function with respect to a scalar variable
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Unformatted text preview: will be another vector. The resulting vector will be directed tangential to the curve that represents the function. Scalar Differentiation in different coordinate systems . An important scalar variable used in differentiations is the time t . Based on the position vector, we can obtain the velocity and acceleration vectors by differentiating the position vectors once and twice, respectively, with respect to time. If the Cartesian coordinates are being used, the axes are independent of time, and differentiation the position vector with respect to time is equivalent to differentiating the individual components with respect to time: v = r = dx i dt x i i...
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