By Charlie Harper
This e-book offers a self-contained remedy of necessary analytic tools in mathematical physics. it truly is designed for undergraduate scholars and it includes good enough fabric for a semester (or 3 area) path in mathematical equipment of physics. With definitely the right collection of fabric, one could use the booklet for a one semester or a one zone direction. the necessities or corequisites are common physics, analytic mechanics, glossy physics, and a operating wisdom of differential an indispensable calculus.
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Extra resources for Analytic Methods in Physics
4 A ) = A - V q 5 + 4 V a A , (b) V x ($A) = q5V x A - A x Vq5, (c) V X V X A = V V . A - V ~ A . 17 By use of Cartesian components, show that (a) V - r = 3, (b) V x r = 0, and (c) Vrn = nrn-2r. 50 CHAPTER 1. 18 Given: A = x2yi + (x - y)k, B = xi, and 4 = xy2z3. 19 If V24 = 0, show that V 4 is both solenoidal and irrotational. 20 If A is irrotational, show that A x r is solenoidal. 21 Find the directional derivative of $(x, y, z) = 2x3 - 3yz at the point (2,1,3) in the direction parallel to the vector with components given by (2,1, -2).
4. INTEGRATION O F VECTOR FUNC"TI0NS B I* = 33 F . dr. 25), dr (tangent to the path) is an element of displacement a t P ( x , y, z). Note that the line integral from B to A is the negative of that from A to B. The w o r k done by a variable force F(x, y, z) in moving an object from A to B is defined as E x a m p l e 15 Calculate the work done by the force F = 2yi along a straight line from A(0, 0,O) to B(2,1,0) rn. Solution: w=L + xyj N in moving an object B F-dr = J(2xi = + xyj) - (dxi + dyj + dzk) l2 xdx + 2 1 y2dy (since the equation for the path is x = 2 ~ ) E x a m p l e 16 Show that the work done on an object of mass m by a net force during a displacement from A to B equals the change in the kinetic energy of the object.
23 Compute the line integral along the line segment joining (O,0,0) and (1,2,4) if A = x 2 i + yj (xx - y)k. A dP +=0 at (Lorentz gauge condition) and show that d2A (a) v 2 A - E o P o - @---- M J and The above equations may be written in the following compact forms q2 A = -poJ and 2 cp =P €0 where the d'Alembertian16 operator O2 is defined by Here Maxwell's equations have been reduced to the study of equations for the vector and scalar potential. 26 By use of Stokes' theorem, prove that V x VV = 0.