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The shell method goes as follows: Consider a volume in three dimensions obtained by rotating a cross-section in the xy-plane around the y-axis. Suppose the cross-section is defined by the graph of the positive function f(x) on the interval [a, b]. Then the formula for the volume will be: ()
In analogy with the cross-section of a solid, the cross-section of an n-dimensional body in an n-dimensional space is the non-empty intersection of the body with a hyperplane (an (n − 1)-dimensional subspace). This concept has sometimes been used to help visualize aspects of higher dimensional spaces. [7]
In physics, the cross section is a measure of the probability that a specific process will take place in a collision of two particles. For example, the Rutherford cross-section is a measure of probability that an alpha particle will be deflected by a given angle during an interaction with an atomic nucleus.
The parallel axis theorem can be used to determine the second moment of area of a rigid body about any axis, given the body's second moment of area about a parallel axis through the body's centroid, the area of the cross section, and the perpendicular distance (d) between the axes. ′ = +
In Cartesian coordinates, the divergence of a continuously differentiable vector field = + + is the scalar-valued function: = = (, , ) (, , ) = + +.. As the name implies, the divergence is a (local) measure of the degree to which vectors in the field diverge.
Image of a helium-4 nucleus; 4 H has a very small cross-section, less than 0.01 barn.. During Manhattan Project research on the atomic bomb during World War II, American physicists Marshall Holloway and Charles P. Baker were working at Purdue University on a project using a particle accelerator to measure the cross sections of certain nuclear reactions.
In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given tensor field along a given curve. For example, the line integral over a scalar field (rank 0 tensor) can be interpreted as the area under the field carved out by a particular curve.
For other objects (for instance, a rolling tube or the body of a cyclist), A may be significantly larger than the area of any cross section along any plane perpendicular to the direction of motion. Airfoils use the square of the chord length as the reference area; since airfoil chords are usually defined with a length of 1, the reference area ...
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