class8 - PHYS 5900 Class 8(Fri Zi-Wei Lin Differentiation...

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PHYS 5900 Class 8 (9/11/2009Fri) Zi-Wei Lin Differentiation In[1]:= ?D D @ f , x D gives the partial derivative f ê x. D @ f , 8 x , n <D gives the multiple derivative n f ë x n . D @ f , x , y , D differentiates f successively with respect to x, y, …. D @ f , 88 x 1 , x 2 , <<D for a scalar f gives the vector derivative H f ê x 1 , f ê x 2 ,… L . D @ f , 8 array <D gives a tensor derivative. à In[2]:= D @ Cos @ 4 x DêH 1 Sin @ 4 x DL ,x D Out[2]= 4 Cos @ 4x D 2 H 1 Sin @ 4x DL 2 4 Sin @ 4x D 1 Sin @ 4x D In[3]:= Simplify @ % D Out[3]= 4 H Cos @ 2x D Sin @ 2x DL 2 In[4]:= D @ 4 x^2 5 x + 8 1 ê x, 8 x, 4 <D Out[4]= 24 x 5 In[5]:= D @ x^3 y^2 2x^2 y + 3 x, x, y D Out[5]= 4x + 6x 2 y In[6]:= ?Dt Dt @ f , x D gives the total derivative df ê dx . Dt @ f D gives the total differential df . Dt @ f , 8 x , n <D gives the multiple derivative d n f ê dx n . Dt @ f , x 1 , x 2 , D gives d ê dx 1 d ê dx 2 f . à
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Example 2.2.6 In z = x 2 y + 3xy 4 , x & y are both a function of t; In[7]:= z = x^2 y + 3 x y^4; In[8]:= Dt @ z, t D Out[8]= 2xyDt @ x, t D + 3y 4 Dt @ x, t D + x 2 Dt @ y, t D + 12 x y 3 Dt @ y, t D In[9]:= % ê . 8 x Sin @ 2t D ,y Cos @ t D< Out[9]= 6 Cos @ t D 4 Cos @ 2t D + 4 Cos @ t D Cos @ 2t D Sin @ 2t D 12 Cos @ t D 3 Sin @ t D Sin @ 2t D Sin @ t D Sin @ 2t D 2 Example 2.2.7 Given the equation of a circle x 2 + y 2 =25, 1) find dy/dx 2) determine the equation of the tangent to the circle at the point (3,4). In[10]:= D @ x^2 + y^2 m 25, x D Out[10]= 2x m 0 Compare the above with In[11]:= Dt @ x^2 + y^2 m 25, x D Out[11]= 2x + 2yDt @ y, x D m 0 In[12]:= Solve @ % ,Dt @ y, x DD Out[12]= :: Dt @ y, x D →− x y >> The slope of the tangent to the circle at the point (3, 4) is In[13]:= slope = Dt @ y, x . % @@ 1 DD ê . 8 x 3, y 4 < Out[13]= 3 4 2 class8.nb
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In[14]:= y y0 m slope H x x0 . 8 x0 3, y0 4 < Out[14]= 4 + y m− 3 4 H 3 + x L In[15]:= % êê Simplify Out[15]= 3x + 4y m 25 Integration In[16]:= ? Integrate Integrate @ f , x D gives the indefinite integral fdx. Integrate @ f , 8 x , x min , x max <D gives the definite integral x min x max fdx. Integrate @ f , 8 x , x min , x max < , 8 y , y min , y max < , D gives the multiple integral x min x max dx y min y max dy … f. à In[17]:= Integrate @ x^4 êH a^2 + x^2 L ,x D Out[17]= a 2 x + x 3 3 + a 3 ArcTan B x a F In[18]:= D @ % ,x Dêê Simplify Out[18]= x 4 a 2 + x 2 In[19]:= Integrate @ 5 x 2 Sqrt @ x D + 32 ê x^3, 8 x, 1, 4 <D Out[19]= 259 6 In[20]:= Integrate @ Sin @ x D ^p, 8 x, 0, Pi <D Out[20]= If B Re @ p D >− 1, π Gamma B 1 + p 2 F Gamma A 1 + p 2 E , Integrate @ Sin @ x D p , 8 x, 0, π < , Assumptions Re @ p D ≤− 1
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This note was uploaded on 04/25/2010 for the course PHYS 5900 taught by Professor Lin during the Fall '09 term at East Carolina University .

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class8 - PHYS 5900 Class 8(Fri Zi-Wei Lin Differentiation...

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