A treatise on the geometry of surfaces by Alfred Barnard Basset

By Alfred Barnard Basset

This quantity is made from electronic pictures from the Cornell collage Library historic arithmetic Monographs assortment.

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Where e (c) A vector A = Ax e1 + Ay e2 + Az e3 can be represented in any of the forms: A = A1 E1 + A2 E2 + A3 E3 A = A1 E 1 + A2 E 2 + A3 E 3 ˆr + Aβ e ˆβ + Az e ˆz A = Ar e depending upon the basis vectors selected . In terms of the components Ax , Ay , Az (i) Solve for the contravariant components A1 , A2 , A3 . (ii) Solve for the covariant components A1 , A2 , A3 . (iii) Solve for the components Ar , Aβ , Az . Express all results in cylindrical coordinates. (Note the components Ar , Aβ , Az are referred to as physical components.

N i are associated constitutes a basis for all third order tensors. Tensor components with mixed suffixes like Cjk with triad basis of the form i Ei Ej Ek C = Cjk where i, j, k have the range 1, 2, . . N. Dyads are associated with the outer product of two vectors, while triads, tetrads,... are associated with higher-order outer products. These higher-order outer or direct products are referred to as polyads. The polyad notation is a generalization of the vector notation. The subject of how polyad components transform between coordinate systems is the subject of tensor calculus.

Let Φ = Φ(r, θ) where r, θ are polar coordinates related to Cartesian coordinates (x, y) by the transformation equations x = r cos θ and y = r sin θ. 1-18 to calculate the Laplacian ∇2 Φ = ∂2Φ ∂2Φ + ∂x2 ∂y 2 in polar coordinates. 13. (Index notation) Let a11 = 3, a12 = 4, a21 = 5, a22 = 6. Calculate the quantity C = aij aij , i, j = 1, 2. 14. Show the moments of inertia Iij defined by (y 2 + z 2 )ρ(x, y, z) dτ I11 = I23 = I32 = − R R 2 I22 = 2 (x + z )ρ(x, y, z) dτ I12 = I21 = − R xyρ(x, y, z) dτ R (x2 + y 2 )ρ(x, y, z) dτ I33 = yzρ(x, y, z) dτ I13 = I31 = − R xzρ(x, y, z) dτ, R xm xm δij − xi xj ρ dτ, where ρ is the density, can be represented in the index notation as Iij = R x1 = x, x2 = y, x3 = z and dτ = dxdydz is an element of volume.

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