Introduction to Mathematical Physics: Methods & Concepts by Chun Wa Wong

By Chun Wa Wong

Mathematical physics presents actual theories with their logical foundation and the instruments for drawing conclusions from hypotheses. advent to Mathematical Physics explains to the reader why and the way arithmetic is required within the description of actual occasions in house. For undergraduates in physics, it's a classroom-tested textbook on vector research, linear operators, Fourier sequence and integrals, differential equations, exact services and capabilities of a posh variable. Strongly correlated with center undergraduate classes on classical and quantum mechanics and electromagnetism, it is helping the coed grasp those useful mathematical talents. It includes complicated themes of curiosity to graduate scholars on relativistic square-root areas and nonlinear structures. It includes many tables of mathematical formulation and references to precious fabrics on the net. It comprises brief tutorials on uncomplicated mathematical issues to aid readers refresh their mathematical wisdom. An appendix on Mathematica encourages the reader to exploit computer-aided algebra to unravel difficulties in mathematical physics.

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It states that the net outflow across a closed surface S is equal to the total divergence in the volume Ω inside S. As a result, we may say that the enclosed divergence “causes” the outflow; that is, the enclosed divergence is a “source” of the outflow. 86) where ε0 is the permittivity of free space. As a result, Ω dτ ρ Q = = ε0 ε0 dσ · E. 87) S This Gauss’s law of electrostatics states that the total flux of the electric field intensity coming out across a closed surface is proportional to the total charge Q enclosed by the surface.

It turns out to be possible to integrate over just the equatorial circle in the xy plane. 76) where n is the normal direction to the spherical surface, as shown in Fig. 7. Hence I= r · n|dσ| = r·n dx dy . k·n The normal n must be calculated from the gradient of the LHS of Eq. 74) ∇r 2 = 2r, or n = er . Hence r · n = a, k · n = z/a, and I=a a dx dy. z 38 Vectors and fields in space y y + Δy 3 D C 4 y 2 A 0 B 1 x x + Δx x Fig. 8 A small rectangular closed loop. When expressed in circular coordinates ρ = (x2 + y2 )1/2 and φ in the xy plane, this simplifies further to a I = 2πa2 −1/2 (a2 − ρ2 ) ρ dρ 0 = 2πa3 .

Hence Γ (∇ × v)z (Δσ)z , = (∇ × v) · Δσ = (∇ × v) · n Δσ, where (Δσ)z = ΔxΔy 50 Vectors and fields in space where n(= k here) is the normal to the surface Δσ enclosed by the loop. 99) is actually independent of the choice of the coordinate system. As a consequence, the same result is obtained whether or not the small loop lies on the xy plane. If we now make the loop infinitesimally small, we obtain the identity 1 Δσ→0 Δσ v · dr = [∇ × v(r)] · n. 100) Thus the curl of a vector field v(r) at a point r in space has the simple interpretation that its component along any direction n is given by the circulation per unit enclosed area of v around an infinitesimally small closed loop surrounding r on a plane perpendicular to n.

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