212HomeWork1sol - Physics 212A Homework#1 Name 1 Practice...

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Unformatted text preview: Physics 212A Homework #1 Name: 1. Practice in transcribing expressions into Mathematica syntax a) Find the determinant of this matrix: ( use Insert, Table/Matrix ) Type it in using Insert Table as per suggestion In[14]:= mat1 = 1 2 3 4 1- 1 1- 1 2 6 7 2 4 1 Out[14]= 88 1, 2, 3, 4 < , 8 1,- 1, 0, 1 < , 8- 1, 2, 6, 7 < , 8 0, 2, 4, 1 << then compute Det In[15]:= Det @ mat1 D Out[15]= 57 First, try to compute y In[16]:= y = Integrate @ Sin @ Sqrt @ t ^ 3 + 1 DD , 8 t, 0, Sqrt @ x D<D Out[16]= x Sin B 1 + t 3 F t It can' t do it. But it can correctly calculate the derivative In[17]:= D @ y, x D Out[17]= Sin B 1 + x 3 2 F 2 x c) Find the first 3 terms in the Taylor's series expansion of about x=0. In[18]:= Series @ Sqrt @ a + Log @ x + 1 DD , 8 x, 0, 3 <D Out[18]= a + x 2 a + H- 1- 2 a L x 2 8 a 3 2 + I 3 + 6 a + 8 a 2 M x 3 48 a 5 2 + O @ x D 4 d) Let A= 1 2 5- 2 4 1- 1 3 and B= 1 1 1- 2 6 4 2 7 . Compute the comutator AB-BA and verify that H AB L T = B T A T and that H A B L- 1 = B- 1 A- 1 Type the matrices in the old fashioed way : In[19]:= A = 88 1, 2, 5 < , 8- 2, 4, 1 < , 8 0,- 1, 3 << Out[19]= 88 1, 2, 5 < , 8- 2, 4, 1 < , 8 0,- 1, 3 << In[20]:= B = 88 1, 1, 1 < , 8- 2, 6, 0 < , 8 4, 2, 7 << Out[20]= 88 1, 1, 1 < , 8- 2, 6, 0 < , 8 4, 2, 7 << Use MatrixForm to make sure we got it right : In[23]:= MatrixForm @ A D Out[23]//MatrixForm= 1 2 5- 2 4 1- 1 3 In[24]:= MatrixForm @ B D Out[24]//MatrixForm= 1 1 1- 2 6 4 2 7 Commutator = In[25]:= A.B- B.A Out[25]= 88 18, 18, 27 < , 8 8, 4, 9 < , 8 14,- 9,- 22 << 2 212HomeWork1sol.nb In[26]:= MatrixForm @ % D Out[26]//MatrixForm= 18 18 27 8 4 9 14- 9- 22 H AB L T = B T A T : In[27]:= TrAB = Transpose @ A.B D Out[27]= 88 17,- 6, 14 < , 8 23, 24, 0 < , 8 36, 5, 21 << In[28]:= TrBTrA = Transpose @ B D .Transpose @ A D Out[28]= 88 17,- 6, 14 < , 8 23, 24, 0 < , 8 36, 5, 21 << In[29]:= TrAB- TrBTrA Out[29]= 88 0, 0, 0 < , 8 0, 0, 0 < , 8 0, 0, 0 << In[30]:= TrAB TrBTrA Out[30]= True H A B L- 1 = B- 1 A- 1 : In[31]:= Inverse @ A.B D- Inverse @ B D .Inverse @ A D Out[31]= 88 0, 0, 0 < , 8 0, 0, 0 < , 8 0, 0, 0 << 212HomeWork1sol.nb 3 In[32]:= Inverse @ A.B D == Inverse @ B D .Inverse @ A D Out[32]= True Use Mathematica to evaluate e) x Tan @ k x D 2 x In[33]:= Integrate @ x Tan @ k x D ^ 2, x D Out[33]=- x 2 2 + Log @ Cos @ k x DD k 2 + x Tan @ k x D k f) ¥ e- s x x 3 Cos @ k x D x for positive s and real k. In[34]:= Integrate A ª- s x x 3 Cos @ k x D , 8 x, 0, ¥ < , Assumptions fi 8 s > 0, k ˛ Reals <E Out[34]= 6 I k 4- 6 k 2 s 2 + s 4 M I k 2 + s 2 M 4 g) ¥ J @ k r D 1 + k 2 r 2 r for positive values of k (look up BesselJ ) In[35]:= Integrate B BesselJ @ 0, k r D 1 + k 2 r 2 , 8 r, 0, ¥ < , Assumptions fi k > F Out[35]= BesselI A 0, 1 2 E BesselK A 0, 1 2 E k 2. Practice using DSolve. The differential equation d n d t =Α n- Β n 2 , is a model for population dynamics....
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This note was uploaded on 12/07/2010 for the course PHYS PHYSICS 21 taught by Professor Alicea,j. during the Fall '10 term at UC Irvine.

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212HomeWork1sol - Physics 212A Homework#1 Name 1 Practice...

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