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OpenStax textbooks follow a traditional peer review process aimed at ensuring they meet a high quality standard before publication. Textbooks are developed and peer-reviewed by educators in an attempt to ensure they are readable and accurate, meet the scope and sequence requirements of each course, are supported by instructor ancillaries, and are available with the latest technology-based ...
2 Differential calculus. 3 Integral calculus. 4 Special functions and numbers. 5 Absolute numerical. ... Volume integral; Jacobian; Hessian; Advanced. Calculus on ...
The substitution is described in most integral calculus textbooks since the late 19th century, usually without any special name. [5] It is known in Russia as the universal trigonometric substitution , [ 6 ] and also known by variant names such as half-tangent substitution or half-angle substitution .
Displacement is the shift in location when an object in motion changes from one position to another. [2] For motion over a given interval of time, the displacement divided by the length of the time interval defines the average velocity (a vector), whose magnitude is the average speed (a scalar quantity).
If one knows that the volume of a cone is (), then one can use Cavalieri's principle to derive the fact that the volume of a sphere is , where is the radius. That is done as follows: Consider a sphere of radius r {\displaystyle r} and a cylinder of radius r {\displaystyle r} and height r {\displaystyle r} .
Now since z 1/2 = e (Log z)/2, on the contour outside the branch cut, we have gained 2 π in argument along γ. (By Euler's identity, e iπ represents the unit vector, which therefore has π as its log. This π is what is meant by the argument of z. The coefficient of 1 / 2 forces us to use 2 π.)
This article summarizes several identities in exterior calculus, ... ( Hodge dual of constant function 1 is the volume form) Co-differential operator properties ...
Integral as area between two curves. Double integral as volume under a surface z = 10 − ( x 2 − y 2 / 8 ).The rectangular region at the bottom of the body is the domain of integration, while the surface is the graph of the two-variable function to be integrated.