mathematical methods for scientists and engineers donald a mcquarrie pdf

... 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 nor polished. I have a “Warnings and Disclaimers” section below that is a little amusing, and an appendix on probability that I feel concisesly captures ... graphics and no examples. There is an exception to this rule: When the title also contains the word Scientists or Engineers ” the advanced book may be quite suitable for actually learning the material. xxvii ...

Ngày tải lên: 06/08/2014, 01:21

... uch that the ordered triple of vectors a, b and n form a right-handed system. 29 a b b θ b Figure 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  is parallel to a. Show that a ì b = a ì b . Finally prove the distributive law for arbitrary b and c. Hint 2.5 Write the vectors in their rectangular components and use, i ... ways of labeling the axes i n a three-dimensional rectangular coordinate system. These are called right-handed and left-handed coordinate systems. See Figure 2.7. Any other labelling of the axes...

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... if u (a) and u(b) are of 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 x 2 − xy + y 2 = 3. Hint, Solution 3.8.5 Maxima and Minima Exercise 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 not simultaneously zero which would produce an indeterminate form....

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... denoted with a definite integral,  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- conquer strategy. 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 ∆x i will be negative. From this observation, it is clear that  b a f(x) dx = −  a b f(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 integral on the right is easier to evaluate than...

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... g  d dt (af) = a  f + af  5.2 Gradient, Divergence and Curl Scalar 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 Integral Exercise 4.1 (mathematica/calculus/integral/fundamental.nb) Evaluate  (2x + 3) 10 dx. Hint, Solution Exercise 4.2 (mathematica/calculus/integral/fundamental.nb) Evaluate  (ln x) 2 x dx. Hint,...

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... u 1 , u 2 and u 3 are real numbers and ı,  and k are objects which satisfy ı 2 =  2 = k 2 = −1, ı = k, ı = −k and the usual associative and distributive laws. Show that for any quaternions ... Euler’s formula. 189 Chapter 6 Complex Numbers I’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 complex number 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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... 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 generalize the 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 substitutions z = x + ıy and z = r e ıθ , respectively. Then we can separate the real and imaginary ... value of the arctangent that is between 0 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 Form We can...

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... 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 that Log 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 b There is not an analogous identity for the principal branch of the logarithm since Arg(ab) is not in general the s ame as Arg (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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... The logarithm does not have a branch point at that point. Since arctan(1/ζ) does not have a branch point at ζ = 0, arctan(z) does not have a branch point at infinity. 2. w = arctanh(z) z = tanh(w) z ... the logarithm term tends to −1 The logarithm does not have a branch point at that point. Since arctanh(1/ζ) does not have a branch point at ζ = 0, arctanh(z) does not have a branch point at infinity. 3. w ... ζ −3/2 has a branch point at ζ = 0, while (1 −ζ 3 ) 1/2 is 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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... ıv(x, y) where u and v are real-valued functions. We equate the real and imaginary parts of Equation 8.1 to obtain another form for the Cauchy-Riemann equations in Cartesian coordinates. u x = v y , ... modulus and argument of this to obtain two equations.) A sufficient condition for analyticity of f(z) is that the Cauchy-Riemann equations hold and the first partial derivatives of φ exist and are continuous ... z 8.3 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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... 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-Riemann equations ... we see that the Cauchy-Riemann equations for à and are satised if and only if the Cauchy-Riemann equations for u and v are satisfied. The continuity of the first partial derivatives of u and v implies ... Solution Cauchy-Riemann Equations Exercise 8.6 If f(z) is analytic in a domain and has a constant real part, a constant imaginary part, or a constant modulus, show that f(z) is constant. Hint, Solution 389 ...

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... equate the real and imaginary parts. u r = 1 r v θ , v r = − 1 r u θ u r = 1 r v θ , u θ = −rv r Solution 8.15 Since w is analytic, u and v satisfy the Cauchy-Riemann equations, u x = v y 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 f 1 (z) and f 2 (z) defined on the domains D 1 and D 2 , ... hyperbolic sine has an essential singularity at infinity, the function has an essential singularity at i nfini ty as well. The point at infinity is a non-isolated si ngularity because there is no...

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... that the integrand is analytic inside 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  C 1 z dz ... 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 products yields, a  x 3 + ı3x 2 y ... the Jacobian of f and g vanishes, then f x g y − f y g x = 0. This is a first order partial differential equation for f that has the general solution f(x, y) = h(g(x, y)). Prove that an analytic...

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... By taking the limit as r → ∞ we see that the modulus of the integral is bounded above by zero. Thus the integral vanishes. Now we assume that f(z) is analytic and evaluate the integral with Cauchy’s ... integral has the value ı2π by the Cauchy-Goursat Theorem. The third integral vanishes by Cauchy’s Theorem as the integrand is analytic inside and on the contour.  C f(z) z 3 dz = ı2π 524 11.1 Cauchy’s ... shows that the value of f(z) and all its derivatives in a domain are determined by the value of f(z) on the boundary of the domain. Consider the first formula of the result, Equation 11.1. We deform...

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... Function. Analyticity. Recall that a sufficient condition for the analyticity of a function f(z) in a domain is that  C f(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 lim n→∞ 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 converges to 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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Ngày tải lên: 06/08/2014, 01:21

Ngày tải lên: 06/08/2014, 01:21

Ngày tải lên: 06/08/2014, 01:21

Ngày tải lên: 06/08/2014, 01:21

Ngày tải lên: 06/08/2014, 01:21