Vector Analysis And the Theory of Relativity
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In ordinary space, i.e., where the a;'s are rectangular Cartesian coordinates and where the y's are orthogonal co- ordinates, /ii = in-' a result which is sometimes useful in the calculation of the coefficients /n, /22, /as • • • of the form (ds)^ in the curvilinear coordinates y.
3. Resolution of tensors If we consider any comriant tensor Xr of rank one and take the inner product of this into a direction tensor Z^""^ we derive the invariant Xrl^^^ (r umbral; Rule (d)). This we call the resolve...d part of the co variant tensor along the direction l^''\ Let us now make a transformation of coordinates from x to y and consider the .coordinate line y^'\ The n components of the direction tensor for this curve are proportional to a^) ir=h--:n) To determine the actual values of these components we must divide through by the positive square root of ^""a^aT^ am umbral) and this is equivalent to A^.
58 VECTOR ANALYSIS AND RELATIVITY The equations defining the covariant correspondence for a tensor of the first rank are Yi = -X'r^ (l = I, ■ ■ ■ , n; r umbral) = A^ times the resolved part of the tensor Xr along the co- ordinate direction y^^^ Example Space polar coordinates y in ordinary space of three dimensions.
What to read after Vector Analysis And the Theory of Relativity?
You can find similar books in the "Read Also" column, or choose other free books by Francis D Francis Dominic Murnaghan to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
In ordinary space, i.e., where the a;'s are rectangular Cartesian coordinates and where the y's are orthogonal co- ordinates, /ii = in-' a result which is sometimes useful in the calculation of the coefficients /n, /22, /as • • • of the form (ds)^ in the curvilinear coordinates y.
3. Resolution of tensors If we consider any comriant tensor Xr of rank one and take the inner product of this into a direction tensor Z^""^ we derive the invariant Xrl^^^ (r umbral; Rule (d)). This we call the resolve...d part of the co variant tensor along the direction l^''\ Let us now make a transformation of coordinates from x to y and consider the .coordinate line y^'\ The n components of the direction tensor for this curve are proportional to a^) ir=h--:n) To determine the actual values of these components we must divide through by the positive square root of ^""a^aT^ am umbral) and this is equivalent to A^.
58 VECTOR ANALYSIS AND RELATIVITY The equations defining the covariant correspondence for a tensor of the first rank are Yi = -X'r^ (l = I, ■ ■ ■ , n; r umbral) = A^ times the resolved part of the tensor Xr along the co- ordinate direction y^^^ Example Space polar coordinates y in ordinary space of three dimensions.
What to read after Vector Analysis And the Theory of Relativity?
You can find similar books in the "Read Also" column, or choose other free books by Francis D Francis Dominic Murnaghan to read onlineMoreLess
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