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mathematical methods for scientists and engineers donald a mcquarrie pdf

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 1 pdf

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 1 pdf

Kĩ thuật Viễn thông

... criticism. You can reach me at sean@caltech.edu.ã Reading this book impairs your ability to drive a car or operate machinery.ã This book has been found to cause drowsiness in laboratory animals.ã This ... Unfortunately, the text is neither complete norpolished. I have a “Warnings and Disclaimers” section below that is a little amusing, and an appendix on probabilitythat I feel concisesly captures ... graphics and no examples. There is an exception to thisrule: When the title also contains the word Scientists or Engineers ” the advanced book may be quite suitable for actually learning the material.xxvii...
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Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 2 ppt

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 2 ppt

Kĩ thuật Viễn thông

... uch thatthe ordered triple of vectors a, b and n form a right-handed system.29 a bbθbFigure 2.14: The vec tor b written as a s um of components orthogonal and parallel to a. and that a × ... orthogonal to a and bisparallel to a. Show that a ì b = a ì b.Finally prove the distributive law for arbitrary b and c.Hint 2.5Write the vectors in their rectangular components and use,i ... ways of labeling the axes i n a three-dimensional rectangular coordinate system. These are calledright-handed and left-handed coordinate systems. See Figure 2.7. Any other labelling of the axes...
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Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 3 pptx

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 3 pptx

Kĩ thuật Viễn thông

... if u (a) and u(b) areof opposite sign then u(x) has at least one zero on the interval (a, b).Maxima and Minima. If u(x) is continuous on [a, b] then u(x) has a maximum and a minimum on [a, b]. ... (mathematica/calculus/differential/implicit.nb)Find y(x) and y(x), given that x2− xy + y2= 3.Hint, Solution3.8.5 Maxima and MinimaExercise 3.15 (mathematica/calculus/differential/maxima.nb)Identify any maxima and minima of the following ... −f (a) g(b) − g (a) .We have assumed that g (a) = g(b) so that the denominator does not vanish and that f(x) and g(x) are notsimultaneously zero which would produce an indeterminate form....
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Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 4 pptx

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 4 pptx

Kĩ thuật Viễn thông

... denoted with a definiteintegral,b a f(x) dx.The area is signed, that is, if f(x) is negative, then the area is negative. We measure the area with a divide -and- conquerstrategy. First partition ... We assume that each of the above integrals exist. If a ≤ b, and we integrate from b to a, then each of the ∆xiwill benegative. From this observation, it is clear thatb a f(x) dx = − a bf(x) ... the formu dv = uv −v du.So what is the usefulness of this? Well, it may happen for some integrals and a good choice of u and v that the integralon the right is easier to evaluate than...
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Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 5 pdf

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 5 pdf

Kĩ thuật Viễn thông

... gddt(af) = a f + af5.2 Gradient, Divergence and CurlScalar and Vector Fields. A scalar field is a function of position u(x) that assigns a scalar to each point in space. A function that gives ... temperature of a material is an example of a scalar field. In two di men sions , you can graph a scalar field as a surface plot, (Figure 5.1), with the vertical axis for the value of the function. A ... Indefinite IntegralExercise 4.1 (mathematica/calculus/integral/fundamental.nb)Evaluate(2x + 3)10dx.Hint, SolutionExercise 4.2 (mathematica/calculus/integral/fundamental.nb)Evaluate(ln x)2xdx.Hint,...
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Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 6 pps

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 6 pps

Kĩ thuật Viễn thông

... u1, u2 and u3are real numbers and ı,  and k are objects which satisfyı2= 2= k2= −1, ı = k, ı = −k and the usual associative and distributive laws. Show that for any quaternions ... Euler’s formula.189 Chapter 6Complex NumbersI’m sorry. You have reached an imaginary number. Please rotate your phone 90 degrees and dial again.-Message on answering machine of Cathy Vargas.6.1 ... This is called the compl ex plane or the Argand diagram. (See Figure 6.2.) A complexnumber written as z = x + ıy is said to be in Cartesian form, or a + ıb form.Recall that there are two ways of...
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Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 7 ppt

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 7 ppt

Kĩ thuật Viễn thông

... real axis and approach infinity via positive real numbers.We could walk along the positive imaginary axis and approach infinity via pure imaginary numbers. We could generalizethe real variable ... write a function of a complex variable z as a function of x and y or as a function of r and θ with the substitutionsz = x + ıy and z = reıθ, respectively. Then we can separate the real and imaginary ... value of the arctangent that is between0 and π. The domain and a plot of the selected values of the arctangent are shown in Figure 7.8.CONTINUE.7.4 Cartesian and Modulus-Argument FormWe can...
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Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 8 ppt

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 8 ppt

Kĩ thuật Viễn thông

... z a = ln |z a | + ı Arg (z a ) , a Log z = a ln |z| + a Arg(z) and Arg (z a ) is not necessarily the same as a Arg(z) we see thatLog z a = a Log z.Consider the logarithm of a product.log(ab) ... |ab| + ı arg(ab)= ln |a| + ln |b|+ ı arg (a) + ı arg(b)= log a + log bThere is not an analogous identity for the principal branch of the logarithm since Arg(ab) is not in general the s ame asArg (a) ... the two equations (for the real and imaginary parts)sin x cosh y = 0 and cos x sinh y = 0.Since cosh is real-valued and positive for real argument, the first equation dictates that x = nπ, n...
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Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 9 ppt

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 9 ppt

Kĩ thuật Viễn thông

... The logarithm does not have a branch point at thatpoint. Since arctan(1/ζ) does not have a branch point at ζ = 0, arctan(z) does not have a branch point atinfinity.2.w = arctanh(z)z = tanh(w)z ... the logarithm term tends to −1 The logarithm does not have a branch point at thatpoint. Since arctanh(1/ζ) does not have a branch point at ζ = 0, arctanh(z) does not have a branch point atinfinity.3.w ... ζ−3/2has a branch point at ζ = 0, while (1 −ζ3)1/2is not singular there. Since f(1/ζ) has a branch point at ζ = 0,f(z) has a branch point at infinity.There are several ways of introducing branch...
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Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 10 doc

Advanced Mathematical Methods for Scientists and Engineers Episode 1 Part 10 doc

Kĩ thuật Viễn thông

... ıv(x, y) where u and v are real-valued functions. We equate the real and imaginary partsof Equation 8.1 to obtain another form for the Cauchy-Riemann equations in Cartesian coordinates.ux= vy, ... modulus and argument of this to obtaintwo equations.) A sufficient condition for analyticity of f(z) is that the Cauchy-Riemannequations hold and the first partial derivatives of φ exist and are continuous ... z8.3 Harmonic Functions A function u is harmonic if its second partial derivatives exist, are continuous and satisfy Laplace’s equation ∆u = 0.2(In Cartesian coordinates the Laplacian is ∆u...
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Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 1 pps

Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 1 pps

Kĩ thuật Viễn thông

... that the imaginary part of f(z) is a constant and conclude that f(z) is constant.Constant Imaginary Part. Next assume that f(z) has constant imaginary part. We solve the Cauchy-Riemannequations ... we see that the Cauchy-Riemann equations for à and are satised if and only if the Cauchy-Riemannequations for u and v are satisfied. The continuity of the first partial derivatives of u and v implies ... SolutionCauchy-Riemann EquationsExercise 8.6If f(z) is analytic in a domain and has a constant real part, a constant imaginary part, or a constant modulus, showthat f(z) is constant.Hint, Solution389...
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Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 2 pptx

Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 2 pptx

Kĩ thuật Viễn thông

... equate the real and imaginary parts.ur=1rvθ, vr= −1ruθur=1rvθ, uθ= −rvrSolution 8.15Since w is analytic, u and v satisfy the Cauchy-Riemann equations,ux= vy and ... removable singularity at z = 3, a pole of order 6 at z = −ı and an essential singularity at z∞.436 Result 9.1.2 Consider analytic functions f1(z) and f2(z) defined on the domains D1 and D2, ... hyperbolic sine has an essential singularity at infinity, the function has an essential singularityat i nfini ty as well. The point at infinity is a non-isolated si ngularity because there is no...
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Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 3 ppt

Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 3 ppt

Kĩ thuật Viễn thông

... that the integrand is analyticinside and on the circle, which is simple and closed. By the Cauchy-Goursat Theorem, the integral vanishes.We cannot apply the Cauchy-Goursat theorem to evaluateC1zdz ... dx≤b a |f(x)||dx| ≤ (b − a) max a x≤b|f(x)|.466 with a, b and c complex-valued constants and d a real constant. Substituting z = x + ıy and expanding productsyields, a x3+ ı3x2y ... the Jacobian of f and g vanishes, thenfxgy− fygx= 0.This is a first order partial differential equation for f that has the general solutionf(x, y) = h(g(x, y)).Prove that an analytic...
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Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 4 ppsx

Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 4 ppsx

Kĩ thuật Viễn thông

... By taking the limit as r → ∞ we see that the modulus of the integral is bounded above by zero. Thus the integralvanishes.Now we assume that f(z) is analytic and evaluate the integral with Cauchy’s ... integral has the value ı2π by theCauchy-Goursat Theorem. The third integral vanishes by Cauchy’s Theorem as the integrand is analytic inside and on the contour.Cf(z)z3dz = ı2π524 11.1 Cauchy’s ... shows that the value of f(z) and all its derivatives in a domain are determined by the value off(z) on the boundary of the domain. Consider the first formula of the result, Equation 11.1. We deform...
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Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 5 pps

Advanced Mathematical Methods for Scientists and Engineers Episode 2 Part 5 pps

Kĩ thuật Viễn thông

... Function.Analyticity. Recall that a sufficient condition for the analyticity of a function f(z) in a domain is thatCf(z) dz = 0 for all simple, closed contours in the domain.Consider a power series ... both 1 and −1.2. There exists a sequence {a n} such that a n> 1 for all n and limn→∞ a n= 1.3. There exists a divergent geometric series whose terms converge.4. There exists a sequence ... whose even terms are greater than 1, whose odd terms are less than 1 and that convergesto 1.5. There exists a divergent series of non-negative terms,∞n=0 a n, such that a n< (1/2)n.6....
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