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Multivariable Calculus Module I: Analytical Geometry and the Gradient Vector

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Multivariable Calculus Module I: Analytical Geometry and the Gradient Vector Multivariable Calculus Module I: Analytical Geometry and the Gradient Vector Matthew F. Causley, Ruben Hayrepetyan, Allan Taylor Kettering University Contents 1 Introduction 2 2 Example: Stable Points in Space between the Sun and Earth2 2.1 Further investigation: Visualizing the Force Field............................3 3 Project: Finding the Other Points4 4 Example: The Hall Effect 5 4.1 the Hall Voltage..............................................6 4.2 Hall Voltage Sensor: a Sample Calculation................................6 5 Project: 7 5.1 The Biot Savart Law............................................7 1 1 Introduction In this module we will use analytical geometry to formulate and solve problems arising in physics and engineering. This requires familiarity with vectors, to measure the direction and magnitudes of fundamental quantities such as distances, velocities and forces. When we measure the rates of change with respect to directional coordinates, the need for a gradient vector naturally arises. Below we will consider a problem in which the gradient vector is used to define critical points in orbital trajectories of satellites. This problem is not only very interesting, but also can be studied without complex calculations. By introducing basic notation along the way, we are able to focus on connecting mathematical operations and physical concepts. 2 Example: Stable Points in Space between the Sun and Earth To conserve fuel, space satellites are often positioned at locations where the gravitational acceleration due to large masses (such as the sun and the earth, see Figure1) is minimal. There exists a point between the two bodies where the accelerations are equal and opposite (commonly called the Lagrangian point L1). We can compute this position in space using calculus. Figure 1: Lagrange Points in the earth-sun plane. The point L1 lies on the line between the two bodies, and indicates where the gravitational accelerations balance out. In the right figure, potential lines are included, indicating the magnitude of gravitational forces. Source: Wikipedia Consider the plane containing both the sun and the earth, with the sun centered at the point S(0; 0; 0), and the earth centered at E(xe; 0; 0). We will examine a satellite with mass m, located at a position (x; y; z). According to Newton's law of gravitation, the forces acting on this point due to the sun and earth respectively, are GMsm GMem Fs = 2 ^r; Fe = 2 ^re; (1) r re where G is the gravitational constant, Ms and Me are the masses of the sun and earth, respectively. The vectors and lengths are hx; y; zi hxe − x; y; zi p 2 2 2 p 2 2 2 ^r = ; ^re = ; r = x + y + z ; re = (xe − x) + y + z : (2) r re Since the point L1 is collinear with the sun and the earth, we can assume it is at (x; 0; 0). Then, ^r = ^ı, ^re = −^ı, and the force becomes a scalar quantity. If we also incorporate centrifugal forces, then the total force at L1 is Ms Me Msx F = Gm − 2 + 2 + 3 : (3) x (xe − x) (xe) 2 Setting the force equal to zero produces a 5th order equation for x, 2 3! GMsm Me x x 2 −1 + + = 0: (4) x Ms xe − x xe But we are only interested in the one satisfying 0 < x < xe. Furthermore, we can reduce the difficulty of the problem by approximating the solution. We observe that two of the quantities are small, namely, x − x x M δ = e = 1 − ; = e : xe xe Ms Then, the terms in parentheses from equation (4) can be written as 1 − δ 2 −1 + + (1 − δ)3 = 0 δ (1 − δ)2 + δ2 (1 − δ)3 = δ2 + δ2 (1 − 3δ) ≈ δ2 ≈ 3δ3: Thus, we find that δ ≈ p3 /3, and in terms of the original variables we have r ! 3 Me x = xe 1 − : (5) 3Ms 2.1 Further investigation: Visualizing the Force Field In order to find the Lagrange point (5), we assumed that the satellite was located at (x; 0; 0), i.e. in the same plane of motion as the sun and the earth. What if we allow the y and z coordinates to vary? Then, the forces will act in different directions. We define the gravitational potential energy functions for the sun and earth, respectively as GMs GMe Vs(x; y; z) = − ;Ve(re) = − ; (6) r re again using the definitions from equation (2). The force acting on the satellite will now be given by a vector F =< Fx;Fy;Fz >, which is given by taking derivatives of the potential functions with respect to each variable @ @ @ F = m (V + V ) ;F = m (V + V ) ;F = m (V + V ) : x @x s e y @y s e z @z s e rather than writing partial derivatives, we can use the gradient vector to define the force vector @ @ @ F = mrV (x; y; z); r = ; ; : @x @y @z Upon evaluating the partial derivatives, we recover precisely equations (1)! For each point (x; y; z) in space, a force given by F(x; y; z) will act on the satellite. How do we visualize this? The Matlab script at the end of this module was used to produce Figure2, which shows the force field (1), with appropriately scaled coefficients. We can visualize the satellite as a marble, placed on the red plane in the Figure. If the marble is too close to either the sun or the earth, the marble will roll into them (retrograde), along paths determined by the gradient. Only at Lagrange points, such as L1, will the marble not roll (where the gradient vanishes). Here, the motion is stable, and so we sometimes call L1 a "stable point". 3 Figure 2: A plot of the force field (1). The satellite will move along the valleys, in the direction of gradients. 1 %Compute and plot the gravitational potential of the earth and sun 2 %Written by Allan Taylor 3 %Kettering University 4 clc; clear; 5 6 %I modified the numbers to make the plot come out nice... 7 Me = 1;%5.972e24*5e6;%mass of earth in kg 8 Ms = 10;%1.989e30*1e2;%mass of sun in kg 9 G = 0.1;%6.673e−11;%gravitational constant inmˆ3/(kg *sˆ2) 10 xe = 2;%1.496e11;%distance from earth to sun inm 11 12 gridx = [−2:0.1:4];%[−2e11:1e10:+4e11]; 13 gridy = [−2:0.1:2];[−2e11:1e10:+2e11]; 14 [x,y] = meshgrid(gridx,gridy); 15 16 %compute distances 17 re = sqrt((x−xe).ˆ2 + (y−0).ˆ2); 18 rs = sqrt((x−0).ˆ2 + (y−0).ˆ2); 19 20 %acceleration of gravity at that point 21 ge = −G*Me./(re.ˆ2); 22 gs = −G*Ms./(rs.ˆ2); 23 24 % cap acceleration of gravity to −10m/secˆ2 25 ge(ge<−10) = −10;% Magic 26 gs(gs<−10) = −10;% Magic 27 28 %Plot the data 29 figure(1); 30 mesh(gridx,gridy,ge+gs) 31 axis tight; 3 Project: Finding the Other Points In order to find the other 4 Lagrange points, the full force balance equation must be investigated. Your project will involve the following tasks. 1. Beginning with equation (6), compute the forces Fs = rVs(x; y; z), and Fe = rVe(x; y; z). 2. The sun and the earth are attracted to each other by a force equal to Fs(xe; 0; 0). Compute this force. 4 2 Meve 3. The centrifugal force of the earth repels it from the sun. This force is of the form Fc(x; y; z) = 2 ^re. xe Compute this force. 4. Because the earth is in a stable orbit, the forces are balanced. Set Fs(xe; 0; 0) = Fc(xe; 0; 0), and solve for the velocity ve of the earth. 5. Compute the orbital period of the earth, T = 2πxe . e ve 6. Now consider a satellite of mass m, located at a Lagrange point L(x; y; z). Both the sun and earth will mv2 exert attractive forces on this point. Furthermore, it will also have a centrifugal force F (x; y; z) = ^r. cs r2 The force balance is Fs(x; y; z) = Fe(x; y; z) + Fcs(x; y; z). Substitute your expressions, and obtain the force balance. Use this to solve for the satellite velocity v. 2πr 7. Compute the orbital period of the satellite, T = v . 8. In order for the point L(x; y; z) to be a Lagrange point, the orbital period must be the same as that of earth. Set T = Te, and establish a relationship between v and ve. 9. Returning to the force balance for the satellite, eliminate ve in favor of v, and then eliminate v using the force balance between the earth and the sun. You should now have an expression involving only x, y, z, the masses of the sun and earth, and the distance xe between them. 10. Obtain approximate solutions (x; y; z), using the procedure resulting in equation (5). Note that this is one of 5 possible solutions, and one of 3 for which y = z = 0. Examples and Applications Look up the relevant values, such as the distance between the earth (or other planets) and the sun, along with their masses. Discuss numerical examples, and describe possible applications of your results. References Include all sources you used. For example: 1. "Lagrangian point." Wikipedia: The Free Encyclopedia. Wikimedia Foundation, Inc. 22 July 2004. Web. 10 Nov. 2016. 2. Koon, W. S.; Lo, M. W.; Marsden, J. E.; Ross, S. D. (2006). Dynamical Systems, the Three-Body Problem, and Space Mission Design. p. 9. 4 Example: The Hall Effect In many electronic systems, it is useful for us to know the electrical current flowing through a particular part of the circuit.
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