Download PDF by Gabrile Lugo: Differential Geometry in Physics (Lecture Notes)

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Consequently, the right hand side also represents an array of functions. In addition, both expressions are linear on X, since by definition ∇X is linear on X. We conclude that the right hand side can be interpreted as a matrix in which each entry is a 1-forms acting on the vector X to yield a function. The matrix valued quantity ω ij is called the connection form . 10 Definition Let ∇X be a Koszul connection and let {ei } be a frame. 18) The Christoffel symbols are the coefficients that give the representation of the rate of change of the frame vectors in the direction of the frame vectors themselves.

65) 32 CHAPTER 2. 1 Frames As we have already noted in Chapter 1, the theory of curves in R3 can be elegantly formulated by introducing orthonormal triplets of vectors which we called Frenet frames. The Frenet vectors are adapted to the curves in such a manner that the rate of change of the frame gives information about the curvature of the curve. In this chapter we will study the properties of arbitrary frames and their corresponding rates of change in the direction of the various vectors in the frame.

59) We also define the source current 1-form J = Jµ dxµ = ρdt + J1 dx1 + J2 dx2 + J3 dx3 . 56) are equivalent to the equations dF d∗F = 0, = 4π ∗ J. 61) 30 CHAPTER 2. DIFFERENTIAL FORMS Proof: The proof is by direct computation using the definitions of the exterior derivative and the Hodge-∗ operator. dF ∂Ex ∂Ex ∧ dx2 ∧ dt ∧ dx1 − ∧ dx3 ∧ dt ∧ dx1 + ∂x2 ∂x3 ∂Ey ∂Ey ∧ dx3 ∧ dt ∧ dx2 + − 1 ∧ dx1 ∧ dt ∧ dx2 − ∂x ∂x3 ∂Ez ∂Ez − 1 ∧ dx1 ∧ dt ∧ dx3 − ∧ dx2 ∧ dt ∧ dx3 + ∂x ∂x2 ∂Bz ∂Bz ∧ dx3 ∧ dx1 ∧ dx2 − ∧ dt ∧ dx1 ∧ dx2 − ∂t ∂x3 ∂By ∂By ∧ dx2 ∧ dx1 ∧ dx3 + ∧ dt ∧ dx1 ∧ dx3 − ∂t ∂x2 ∂Bx ∂Bx ∧ dx1 ∧ dx2 ∧ dx3 .

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Differential Geometry in Physics (Lecture Notes) by Gabrile Lugo

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