Problem 3.5 Prove the Schwarz inequality (Equation 3.27). Hint: Let ly) IB)-((a\B)/(ala))|a), and use (yly) ≥ 0.
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Q: *Problem 4.34 (a) Apply S_ to |10) (Equation 4.177), and confirm that you get /2h|1–1). (b) Apply S+…
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- Describe all vectors in span{(3,0,2), (-2,0,3)} (so computationally what do the vectors look like?). Also give a geometric description for these vectors (what space are you in and visually what do you get? Be as descriptive as you can!).Example of numerical instability: Take y′ = −5y, y(0) = 1. We know that the solution should decay to zero as x grows. Using Euler’s method, start with h= 1 and compute y1, y2, y3, y4 to try to approximate y(4). What happened? Now halve the interval. Keep halving the interval and approximating y(4) until the numbers you are getting start to stabilize (that is, until they start going towards zero).Problem 2.34 Consider the "step" potential:53 V (x) = [0, x ≤0, Vo, x > 0. (a) Calculate the reflection coefficient, for the case E Vo
- Problem 4.16 It is desired to find the equation for the shortest distance be- tween two points on a sphere. Determine the functional for this problem. (Use spherical coordinates.)Problem 4.25 If electron, radius [4.138] 4πεmc2 What would be the velocity of a point on the "equator" in m /s if it were a classical solid sphere with a given angular momentum of (1/2) h? (The classical electron radius, re, is obtained by assuming that the mass of the electron can be attributed to the energy stored in its electric field with the help of Einstein's formula E = mc2). Does this model make sense? (In fact, the experimentally determined radius of the electron is much smaller than re, making this problem worse).Prove that ||A + B|| ≤ ||A|| + ||B||. This is called the triangle inequality; in twoor three dimensions, it simply says that the length of one side of a triangle ≤sum of the lengths of the other 2 sides. Hint: To prove it in n-dimensional space, write the square of the desired inequality using (10.2) and also use the Schwarz inequality (10.4). Generalize the theorem to complex Euclidean space by using (10.7) and (10.9).
- Figure 1.52 shows a spherical shell of charge, of radiusa and surface density σ, from which a small circular piece of radius b << a has been removed. What is the direction and magnitude of the field at the midpoint of the aperture? Solve this exercise using direct integration.Bit confused you say with is the definition of a complex conjugate but all I've ever seen is |X|^2=(X*)(X). Can you provide maybe a reference or proof of this?1.3. Determine an orthogonal basis for the subspace of C (-1, 1] spanned by functions: {f(x) = x, f(x) = x³, f(x) = x³] using Gram-Schmidt process.