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Difficult integrals may also be solved by simplifying the integral using a change of variables given by the corresponding Jacobian matrix and determinant. [1] Using the Jacobian determinant and the corresponding change of variable that it gives is the basis of coordinate systems such as polar, cylindrical, and spherical coordinate systems.
Thus, in our example expression, the problem is how to absorb the coefficient of the cross-term 8xy into the functions u and v. Formally, this problem is similar to the problem of matrix diagonalization, where one tries to find a suitable coordinate system in which the matrix of a linear transformation is diagonal. The first step is to find a ...
A point P has coordinates (x, y) with respect to the original system and coordinates (x′, y′) with respect to the new system. [1] In the new coordinate system, the point P will appear to have been rotated in the opposite direction, that is, clockwise through the angle . A rotation of axes in more than two dimensions is defined similarly.
Simplification is the process of replacing a mathematical expression by an equivalent one that is simpler (usually shorter), according to a well-founded ordering. Examples include:
In the second step, the distributive law is used to simplify each of the two terms. Note that this process involves a total of three applications of the distributive property. In contrast to the FOIL method, the method using distributivity can be applied easily to products with more terms such as trinomials and higher.
and either y = 0 or z 2 = 1 so that z = ±1. Their two external intersections are x = y, z = 1; x = −y, z = −1. Likewise, the other external intersections are x = z, y = 1; x = −z, y = −1; y = z, x = 1; y = −z, x = −1. Let us see the pieces being put together. Join the paraboloids y = xz and x = yz. The result is shown in Figure 1 ...
Intuitively, a reduction order R relates two terms s and t if t is properly "simpler" than s in some sense.. For example, simplification of terms may be a part of a computer algebra program, and may be using the rule set { x+0 → x, 0+x → x, x*0 → 0, 0*x → 0, x*1 → x, 1*x → x}.
The following exposition assumes that the numbers are broken into two-digit pieces, separated by commas: e.g. 3456 becomes 34,56. In general x,y denotes x⋅100 + y and x,y,z denotes x⋅10000 + y⋅100 + z, etc. Suppose that we wish to divide c by a, to obtain the result b. (So a × b = c.)