Mathematical Tools for Physics - Department of Physics - University

12—Tensors 314 The velocity vector is d~r dt =~e 1 dr dt +~e 2 dφ dt and Eq. (12.40) is df dt = ∂f ∂xi dxi dt = (∇f).~v This sor ...

12—Tensors 315 points,∆x^1 =one cm, sodx^1 /dt= 1cm/sec. The speed however, iscscαcm/sec because the distance moved is greater b ...

12—Tensors 316 The way that the scalar product looks in terms of these bases, Eq. (12.33) is ~v.gradf=~ei dxi dt .~ej(gradf) j=v ...

12—Tensors 317 Metric Tensor The simplest tensor field beyond the gradient vector above would be the metric tensor, which I’ve b ...

12—Tensors 318 Take the case for whichf=yk, then ∂f ∂yj = ∂yk ∂yj =δjk which gives ~e′k=~ei ∂yk ∂xi (12.53) The transformation m ...

12—Tensors 319 The basis vectors associated with this coordinate system point along the directions of the axes. This manifold is ...

12—Tensors 320 ~e′ 0 =~ei ∂xi ∂x′^0 =~e 0 1 √ 1 −v^2 /c^2 +~e 1 v/c √ 1 −v^2 /c^2 ~e′ 1 =~ei ∂xi ∂x′^1 =~e 0 v/c √ 1 −v^2 /c^2 + ...

12—Tensors 321 Exercises 1 On the three dimensional vector space of real quadratic polynomials inx, define the linear functional ...

12—Tensors 322 Problems 12.1 Does the functionTdefined byT(v) =v+cwithca constant satisfy the definition of linearity? 12.2 Let ...

12—Tensors 323 basis, in its reciprocal basis, and mixed (A~in one basis andB~ in the reciprocal basis). A fourth way uses thexˆ ...

12—Tensors 324 using thexˆ,yˆ,zˆbasis. Ans:∂Tij/∂xjin coordinate basis. 12.23 ComputedivT in cylindrical coordinates using both ...

Vector Calculus 2 . There’s more to the subject of vector calculus than the material in chapter nine. There are a couple of type ...

13—Vector Calculus 2 326 Do the simplest example first. What is the circumference of a circle? Use the parametrization x=Rcosφ, ...

13—Vector Calculus 2 327 Suppose that this particle starts at rest fromy= 0, thenE= 0andv= √ 2 gy. Does the total time to reach ...

13—Vector Calculus 2 328 The basic idea is a combination of Eqs. (13.1) and (13.2). Divide the specified curve into a number of ...

13—Vector Calculus 2 329 Gradient What is the line integral of a gradient? Recall from section8.5and Eq. (8.16) thatdf= gradf.d~ ...

13—Vector Calculus 2 330 ˆr θ θˆ Example Verify Gauss’s Theorem for the solid hemisphere,r≤R, 0 ≤θ≤π/ 2 , 0 ≤φ≤ 2 π. Use the vec ...

13—Vector Calculus 2 331 top area. For small enough values of these dimensions, the right side of Eq. (13.18) is simply the valu ...

13—Vector Calculus 2 332 side from the other.* From here on I’ll imitate the procedure of Eq. (13.14). Divide the surface into a ...

13—Vector Calculus 2 333 I need only therˆcomponent of the curl because the surface integral uses only the normal (ˆr) component ...