Download e-book for iPad: Calculus Reference (SparkCharts) by SparkNotes Editors

By SparkNotes Editors

ISBN-10: 1586638963

ISBN-13: 9781586638962

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SparkChartsTM—created by way of Harvard scholars for college kids everywhere—serve as research partners and reference instruments that disguise a variety of university and graduate tuition topics, together with company, machine Programming, medication, legislation, international Language, Humanities, and technology. Titles like tips on how to learn, Microsoft notice for home windows, Microsoft Powerpoint for home windows, and HTML provide you with what it takes to discover luck at school and past. Outlines and summaries conceal key issues, whereas diagrams and tables make tough ideas more straightforward to digest.

This two-page chart experiences derivatives and differentiation, integrals and integration. It additionally covers purposes in:

Geometry
Motion
Probability and statistics
Microeconomics
Finance

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Extra info for Calculus Reference (SparkCharts)

Example text

Then clearly X1 is linearly independent of X2 … Xn , if it cannot be written as a linear superposition of them. g. mutually perpendicular vectors in a plane; obviously one of them cannot be made a multiple of the other (which is the simplest kind of linear superposition) no matter what coefficient is tried. cn are scalars), only occurs when c= c= c=  = c= 0 1 2 3 n It is not hard to find out if this happens or not for a given set of vectors. We just put the X-vectors together as the columns of a matrix, and multiply that with the column vector formed from the c’s.

The most basic example of a geometrical vector is the position vector. e. mutually perpendicular X- and Y-axes. e. the x- and y-coordinates of the point, is its position vector. The same idea can be applied to a point in 3 dimensional space, where the position vector has 3 elements. There are many other examples of physically measurable quantities that are also mathematically represented as vectors, like velocity, acceleration, electric and magnetic fields, etc. So how do matrices fit into the picture?

A shortcut to do this is to set up a new matrix where we can do it all together. We augment the original matrix A with the original column vector of constants to create an augmented matrix a b d e   g h c f i p q  r  and then we perform elementary row operations on the whole (now non-square) augmented matrix to get the identity matrix (3 x 3) in the left hand part of the matrix and the solutions in the right hand side of the matrix. 1 0 0 0 1 0  0 0 1 x y  z  These ideas show that if the matrix of coefficients is non-singular, we can use elementary row operations to invert it and hence to solve equations that have non-zero terms p,q and r on the right hand side.

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Calculus Reference (SparkCharts) by SparkNotes Editors


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