Mathematical Tools for Physics - Department of Physics - University
9—Vector Calculus 1 234 whereu=asinhθ. Put this back into the original variables and you have φ=−Gλ [ sinh−^1 ( L−z √ x^2 +y^2 ) ...
9—Vector Calculus 1 235 Problems 9.1 Use the same geometry as that following Eq. (9.3), and take the velocity function to be~v = ...
9—Vector Calculus 1 236 9.13 Mimic the calculation in section9.6for the divergence in cylindrical coordinates, computing the cur ...
9—Vector Calculus 1 237 (b) The total energy of this electric field is the integral over all space of the energy density. What i ...
9—Vector Calculus 1 238 gravity isenergynot just mass! This energy that you just computed from the gravitational field is then t ...
9—Vector Calculus 1 239 9.33 Theorem: Given that a vector field satisfies∇×~v= 0everywhere, then it follows that you can write~v ...
9—Vector Calculus 1 240 9.42 Translate the preceding vector fields into polar coordinates, then take their divergence and curl. ...
9—Vector Calculus 1 241 Look for a solution for the magnetic field in the formB~ =ˆzBz(r). What is the total current per length ...
Partial Differential Equations . If the subject of ordinary differential equations is large, this is enormous. I am going to exa ...
10—Partial Differential Equations 243 and in the limit this is ∂T ∂t = κ cρ ∂^2 T ∂x^2 (10.3) I was a little cavalier with the n ...
10—Partial Differential Equations 244 If you want to find out if you have a solution, plug in: ∂T ∂t = κ cρ ∂^2 T ∂x^2 is df dt ...
10—Partial Differential Equations 245 functions ofxthat occur whenα > 0 , you will need the circular functions ofx, sines and ...
10—Partial Differential Equations 246 The curves show the temperature dropping very quickly for points near the surface (x= 0orL ...
10—Partial Differential Equations 247 The further condition is that atx= 0the temperature isT 1 e−iωt, so that tells you thatB=T ...
10—Partial Differential Equations 248 Use the same method used before for heat flow in one dimension: separation of variables. A ...
10—Partial Differential Equations 249 (IfA= 0everything is zero, so it has to beD.) You can now write a general solution that sa ...
10—Partial Differential Equations 250 This simply says that the second derivative with respect toyvanishes, so the answer is the ...
10—Partial Differential Equations 251 position. x= 0 x=a y=b y=b−a T(x,b−a)≈ 4 π T 0 ∑∞ `=0 1 2 `+ 1 e−(2`+1)πsin (2`+ 1)πx a Fo ...
10—Partial Differential Equations 252 and this sum does not converge.I’m going to push ahead anyway, temporarily pretending that ...
10—Partial Differential Equations 253 O 0 b y 0 F 0 a 0 x −F 0 y/κ O b y 0 a −F 0 y/κ x If I can find a solution that equals−F 0 ...
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