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In particular, the fundamental theorem of calculus is the special case where the manifold is a line segment, Green’s theorem and Stokes' theorem are the cases of a surface in or , and the divergence theorem is the case of a volume in . [2] Hence, the theorem is sometimes referred to as the fundamental theorem of multivariate calculus.
An illustration of Stokes' theorem, with surface Σ, its boundary ∂Σ and the normal vector n.The direction of positive circulation of the bounding contour ∂Σ, and the direction n of positive flux through the surface Σ, are related by a right-hand-rule (i.e., the right hand the fingers circulate along ∂Σ and the thumb is directed along n).
When the outer Soddy circle has negative curvature, its center is the isoperimetric point of the triangle: the three triangles formed by this center and two vertices of the starting triangle all have the same perimeter. [4] Triangles whose outer Soddy circle degenerates to a straight line with curvature zero have been called "Soddyian triangles ...
In order to apply this to the Navier–Stokes equations, three assumptions were made by Stokes: The stress tensor is a linear function of the strain rate tensor or equivalently the velocity gradient. The fluid is isotropic. For a fluid at rest, ∇ ⋅ τ must be zero (so that hydrostatic pressure results).
The area of a triangle can be demonstrated, for example by means of the congruence of triangles, as half of the area of a parallelogram that has the same base length and height. A graphic derivation of the formula T = h 2 b {\displaystyle T={\frac {h}{2}}b} that avoids the usual procedure of doubling the area of the triangle and then halving it.
The Stokes I, Q, U and V parameters. The Stokes parameters are a set of values that describe the polarization state of electromagnetic radiation.They were defined by George Gabriel Stokes in 1851, [1] [2] as a mathematically convenient alternative to the more common description of incoherent or partially polarized radiation in terms of its total intensity (I), (fractional) degree of ...
In the third picture, this is shown by the coefficients r' and t' (for reflection and transmission of the reversed fields). Everything must interfere so that the second and third pictures agree; beam x has amplitude E and beam y has amplitude 0, providing Stokes relations. The most interesting result here is that r=-r’. Thus, whatever phase ...
The Nagel triangle or extouch triangle of is denoted by the vertices , , and that are the three points where the excircles touch the reference and where is opposite of , etc. This T A T B T C {\displaystyle \triangle T_{A}T_{B}T_{C}} is also known as the extouch triangle of A B C {\displaystyle \triangle ABC} .
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