Mathematical Tools for Physics
8—Multivariable Calculus 233 The quantityλ 2 is usually denotedβin this type of problem, and it is related to temperature byβ= 1 ...
8—Multivariable Calculus 234 Now step up to three dimensions and again place yourself at the origin. This time place a sphere of ...
8—Multivariable Calculus 235 Cross Section, Scattering There are many types of cross sections besides absorption, and the next s ...
8—Multivariable Calculus 236 The cross section for being sent in a direction between these two angles is the area of the ring:dσ ...
8—Multivariable Calculus 237 transmitted and reflected at each surface. b β β β α α α α θ sinβ=nsinα θ= (β−α) + (π− 2 α) + (β−α) ...
8—Multivariable Calculus 238 The derivativedb/dθ= 1 / [dθ/db]. Compute this. dθ db = 2 √ R^2 −b^2 − 4 √ n^2 R^2 −b^2 (33) In the ...
8—Multivariable Calculus 239 At(b 0 ,θ 0 ), Eq. ( 33 ) gives zero and Eq. ( 32 ) tells youθ 0. The coefficientγcomes from the se ...
8—Multivariable Calculus 240 only a few percent of the light is reflected and the rest goes through. The cross section should be ...
8—Multivariable Calculus 241 Problems 8.1 Letr= √ x^2 +y^2 ,x=Asinωt,y=Bcosωt. Use the chain rule to compute the derivative with ...
8—Multivariable Calculus 242 8.8 When currentI flows through a resistanceRthe heat produced isI^2 R. Two terminals are connected ...
8—Multivariable Calculus 243 8.12 Repeat the preceding problem for the drumhead mode of problem 10. The exact result, calculated ...
8—Multivariable Calculus 244 8.22 Taylor’s power series expansion of a function of several variables was discussed in section2.5 ...
8—Multivariable Calculus 245 8.27 Compute the area of an ellipse having semi-major and semi-minor axesaandb. Compare your result ...
8—Multivariable Calculus 246 8.33 Find the gradient ofV, whereV = (x^2 +y^2 +z^2 )e− √ x^2 +y^2 +z^2. 8.34 A billiard ball of ra ...
8—Multivariable Calculus 247 8.44 What is the shortest distance from the origin to the plane defined byA~.(~r−~r 0 ) = 0? Do thi ...
Vector Calculus 1 The first rule in understanding vector calculus isdraw lots of pictures. This subject can become rather abstra ...
9—Vector Calculus 1 249 A h v∆t α ˆn α The area of a parallelogram is the length of one side times the perpendicular distance fr ...
9—Vector Calculus 1 250 each of these areas the fluid has a velocity~vk. This may not be a constant, but as usual with integrals ...
9—Vector Calculus 1 251 Now to implement the calculation of the flow rate: Divide the area intoN pieces of length∆kalong the sla ...
9—Vector Calculus 1 252 y x θk+1 nˆk b θk ˆy ˆx y b/ 2 (b/2) sinθ The velocity field is the same as before,~v(x,y,z) =v 0 ˆxy/b, ...
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