The value of cos(4M) where M is the magnitude of the vector field with potential ƒ = e² sin(лy) cos(π²) at x = 1, y = 1/4, z = 1/3 is 0.602 -0.323 0.712 -0.816 0.781 0.102 0.075 0.013
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- Let A be the vector potential and B the magnetic field of the infinite solenoid of radius R. Then 0 ={8/ B(r) = where r = √x² + y2 is the distance to the z-axis and B is a constant that depends on the current strength I and the spacing of the turns of wire. The vector potential for B is -{} J A(r) = if r> R Bk if r R r< R SThe motion of a point on the circumference of a rolling wheel of radius 4 feet is described by the vector function r(t) = 4(12t - sin(12t))i + 4(1 − cos(12t))j Find the velocity vector of the point. v(t) = Find the acceleration vector of the point. a(t) = Find the speed of the point. s(t) = =Let r(t)=(10 cos(t), 1, 10 sin(t)). Find the unit tangent vector u drawn to point P-(6,1,8). Sketch curve and tangent.
- Sketch the curve with the vector equation r(t) = cos(t)i − cos(t)j + sin(t)k. Show the direction of increasing t with an arrow drawn on your curve.A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by v = (x - y, z + y + 9, z?) and the net is decribed by the equation y = V1 - x² – z7, y > 0, and oriented in the positive y- direction. (Use symbolic notation and fractions where needed.) v · dS = 10n IncorrectFind a equation vector and the equation of the tangent line at the point P0 where t=0.2 on the graph of the vector function r(t)=e2ti+(t2-t)j+(ln(t))k
- (1n |t – 1], e', vî ) 1. Let 7(t) = (a) Express the vector valued function in parametric form. (b) Find the domain of the function. (c) Find the first derivative of the function. (d) Find T(2). (e) Find the vector equation of the tangent line to the curve when t=2. 2. Complete all parts: (a) Find the equation of the curve of intersection of the surfaces y = x? and z = x3 (b) What is the name of the resulting curve of intersection? (c) Find the equation for B the unit binormal vector to the curve when t= 1. Hint: Instead of using the usual formula for B note that the unit binormal vector is orthogonal to 7 '(t) and 7"(t). In fact, an alternate formula for this vector is ア'(t) × ア"(t) ア(t) ×デ"(t)| B(t) =Suppose a particle, whose initial position is (1, 0, 0), moves with velocity given by v(t) = (-1, cos(t), - sin(t)). Compute the vector-valued function that represents the particle's position at any time t = [0, 2π].A net is dipped in a river. Determine the flow rate of water across the net if the velocity vector field for the river is given by v = (x – y, z + y + 9, z?) and the net is decribed by the equation y = V1- x2 - z?, y > 0, and oriented in the positive y- direction. (Use symbolic notation and fractions where needed.) v • dS = Incorrect