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Ch 1: Introduction to vectors, scalar and vectors, dot and cross product, polar and axial vectors; cyclic nature of triple product, Kronecker’s delta and Levi-Civita tensor, linear independence and Wronskian. Appendix C: Linear second order differential equations

Ch.1 Vector calculus: del operator, gradient, div and rot operations, Gauss and Stoke’s theorems. Appendix F: Coordinate systems

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1. Inhomogeneous Second-Order Differential Equations (35 points total) Resistor R = 20.0 Ω, self-inductance L = 80.0 mH, uncharged capacitor C = 200 μF and open switch S S are connected in series to a battery of potential difference V = 120 V. At t = 0 the switch is closed, V C dq so qo = 0 C and Io = 0 A. Also: I = dt For an LRC circuit (shown to right), Kirchhoff’s L second law (around any closed circuit: ΣVi = 0) produces an inhomogeneous second-order differential equation for charge q: Above: An LRC circuit connected in series 2 to battery V and switch S. dq q d q L 2 + R + =V dt C dt Put this equation into the right form and then, using the given numbers and the technique of appendix C, find q(t) and I(t), the general solution for this equation. As in all secondorder differential equations, there will be two "undetermined" constants in the solution; evaluate them using the initial conditions. Your answer should not include an undetermined constant. NOTE: Watch out for units! All units must be in the same system! (35 points) 2. Using components, prove the "BAC-CAB Rule" (10 points) Theorem: For any vectors A, B and C: A x (B x C) = B(A • C) - C(A • B) HINT: Problems like this are simplified by rotating the axes so that: ➢ The x-axis (and i unit vector) point along one of the three vectors ➢ Since two vectors define a plane: you can make the second vector in the x-y plane so it has only i and j components. ➢ Now, the third vector can be in any direction, and has three components. 3. Divergence and Curl in Cartesian Coordinates (20 points total) a) If A = xyzi + x2zj – y2zk, find: i.  • A (the divergence of A) at (4,3,2) ii.  x A (the curl of A) at (4,3,2) (5 points) (5 points) b) Prove that, for any scalar function Φ, the curl of the gradient of Φ is always zero:  x (  Φ) = 0 (10 points) 4. de Moivre’s Theorem (20 points total) a) Use MacLaurin series to prove de Moivre’s theorem (10 points): einx = cos(nx) + i sin(nx) = (cosx + isinx)n HINT: First prove Euler's identity: eix = cos(x) + i sin(x) easily proved. Be sure to prove both parts. Now, the rest of the theorem is b) Use de Moivre’s theorem to find an expression for sin(4x) and cos(4x) in terms involving powers of sin(x) and cos(x). (5 points each, 10 points total) NOTE: Do not spend time simplifying your results. y 5. Gauss’ Theorem (25 points total) Consider a box 8 cm high, 6 cm wide and 4 cm deep (see diagram to right). Water enters the top surface (4 cm x 6 cm) 8 cm at vo = -60j cm/s, and leaves through two surfaces: - through the front (6 cm x 8 cm) at v = 12k cm/s 6 cm - through the right-hand side (4 cm x 8 cm) at v = 48i Assume: - The water is completely incompressible z - All water enters and exits each surface uniformly, and along the normal to the surfaces Remember: the direction of the surface vector S is always outward. 4 cm a) Calculate the net outward flow of fluid twice: (10 points each, 20 points total) i. First, use the left-hand side of Gauss’ Law: v • dS, the sum of this dot product over the areas.  ii. Now, use the right-hand side of Gauss’ Law:  • vdV, the sum of the divergence of the velocity vector over the volume. NOTE: Since the flow through all surfaces is uniform, you do not have to calculate any integrals here! The first integral is a simple sum. For the second: you must first find an expression for  • V using deltas (Δvx, for example), and then find the total. b) Discuss the meaning of the number you got, and its sign. (5 points) NOTE: With divergence, you must always ask the question: is there a source, a sink, or neither? x
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solution of non-homogenous differentia...


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