By Michael J. Crowe
Concise and readable, this article levels from definition of vectors and dialogue of algebraic operations on vectors to the idea that of tensor and algebraic operations on tensors. It also includes a scientific research of the differential and crucial calculus of vector and tensor capabilities of house and time. Worked-out difficulties and ideas. 1968 version
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Extra resources for A history of vector analysis : the evolution of the idea of a vectorial system
38) holds for a weak relative minimum (and hence, for a strong relative or global minimum). 38). A complete proof of this result is too technical for our purpose. Instead, we refer the reader to Ewing [12; p. 45]. Similar technical problems are handled in Sagan [44; p. 82] where he "smooths corners". Briefly, if we write a Taylor series expansion for J(x, Yo(x), u) about u = yb(x), we have J(x,yo(x),u) = J(x,yo(x),yb(x)) + (u - yb(x))Jyl(x,yo(x),yb(x)) + ~ (u - Yb(x))2 Jylyl (x, Yo(x), Yb(x) + O(u - Yb(x))) where 0 < 0 < 1.
Quick d calculations lead to fx = fy = 0, fy, = ~ and finally, that dxfy' = fy I implies that y' = c. Thus, y(x) = cx + Cl so that the boundary conditions y(O) = 1, y(2) = 3 lead to Yo(x) = x + l. 6). Thus, , fy = 0, fy, = ~, fyy' = fy,y = 0 and fy'Y' = (1 1 +y'2 + y'2)-3/2 so that J'(yO, z) = 102(fyZ + fy'Z')dx = 102 ~z'dX and I"(yo, z) = 102(fyyZ2 + 2fyy'ZZ' + fY'y'Z'2)dx = 102 2- 3/ 2Z'2 dx. : 0 for all piecewise smooth functions z(x). 7a) and the condition J'(yo, z) = 0 for all piecewise smooth functions in this example.
There are now many more necessary and sufficient conditions, the technical details are more complex and the topology for possible solutions is more varied. In addition, there are difficulties about whether our problem is well-posed which involve the questions of existence, uniqueness and continuous dependence on the given initial or boundary conditions of the solution. 1 by using Taylor series expansions associated with I(y). The major result is that a critical point solution yo(x) satisfies the Euler-Lagrange equation, which is a second order ordinary differential equation.