(a) Show that when Laplace’s equation
(b) Show that when Laplace’s equation is written in spherical coordinates, it becomes
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- 8. A line initiated from (1,0,2) and parallel to a vector i+j+k what is the parametric equation of the line (if t is a parameter)? A. x=1+ 2i, y= 2j , z= 2+j B. x=1+ 2t, y= 2t C. x=1+ t, y=t, z= 2+t D. x=2+ i, y= 1, z= 2+t 9. Find , dA, where r is a radius of a circle 1e30 Jr=0 В. п/2 С. 4л D. т/4 А. 2л 10. Which of the following is not a function of f: R? → R? A. f(r, y) = ey B. f(r, y) = 1 C. f(r, y) = e +1 D. f(r) = e"arrow_forward3. Find a vector function that represents the curve of intersection of the paraboloid z = r+y and the cylinder r+y = 16.arrow_forwardWhat is the parametric equation of line through the point (1,2, –1) and perpendicular to the plane x-2y+3: =1 O a. t-1=x, 2t3y, -3t=z O b. 1+2t=x, 1-t/23DY, 3t=z O c. 1-t=x, -2+t=Dy, 3-t3z O d. 1+t=x, 2-2t3Dy, 3t-1=z O e. t=x, 2t-23y, 3t+1=zarrow_forward
- (a) Find a vector function that describes the curve of the intersection of the cylinder z+y+2z=4. 2² +y² = 4 and the plane (b) Use the vector function to find the parametric equations for the tangent line to this curve at t = π/6.arrow_forwardthe vector function r ( t ) = ⟨ 6 cos ( t ) , 3 − 3 sin ( t ) , 2 + 3 sin ( t ) ⟩ 1.Find the equation of the line tangent to the space curve at ( 0 , 2 , 3 ) 2.Find all the points where the space curve intersects the plane z=7.arrow_forwarda) A three dimensional motion of an object is given by the vector function r(t) = 4 cos t i+ 4 sin tj+5 k. Sketch the motion of the object when 0arrow_forwardFind a scalar equation of the plane that passes through the point P(1, 1, 1) and contains the line with parametric equations x = 1 + t y = 3t − 1 z = t .arrow_forwardFind the vector equation that represents the curve of intersection of the cylinder z + y = 16 and the surface z = ze". Write the equation so the r(t) term contains a cos(t) term. z(t) %3D y(t) z(t) =arrow_forward1. A body rotates with a constant angular velocity w = w,î + wzĵ + w3k about an axis. Let ř be the position vector of a point P on the body measured from the origin. Then the linear velocity i of the rotation is i = w x i if ř = xî + yî + zk. Show that w =- curl v . 2arrow_forward5. a) Find parametric equations for the tangent line to the curve of intersection between the paraboloid z = x²+y² and the ellipsoid 4x² +y²+z² = 9 at the point (-1,1,2). Also, find a vector function that represents the curve of intersection. b) Find parametric equations for the line tangent to the curve of intersection between S₁: x² + y² = 4 and S₂ : x² + y² - z = 0 at the point (√2, √2,4). Find a vector function for the curve of intersection.arrow_forwardAssume that an object is moving along a parametric curve and the three vector function. T (t), N(t), and B (t) all exist at a particular point on that curve. CIRCLE the ONE statement below that MUST BE TRUE: (a) B. T=1 (b) T x B = N (B is the binormal vector.) v (t) (c) N (t) = |v (t)| (d) N (t) always points in the direction of velocity v (t). (e) a (t) lies in the same plane as T (t) and N (t).arrow_forwardFind the vector equation that represents the curve of intersection of the cylinder x² + y² 4 and the surface z = x+4y. = Write the equation so the x(t) term contains a cos(t) term. x(t) = y(t) = z(t) =arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage