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0 is the initial position vector and !v 0 is the constant velocity vector of the object. b.) You have entered an incorrect email address! A thin wire has the shape of the parabola y = x^2 from the origin to the point (1,1) and the line segment from (1. Determine whether or not the field \vec F(x, y) = 2x \cos (3y)i -3x^2 \sin (3y) j is conservative. Evaluate the integral of F along the following path. Evaluate C ( 2 x y ) d x + ( x + 2 y ) d y . div F = _____. SCALAR FIELDS AND VECTOR FIELDS 169 Chapter 22. dr, where F = (4 sin x, 4 cos y, 10 x z) and C is the path given by r(t) = (-2t^3, 3t^2, -2t) for 0 less than or equal to t less than or equal to 1. Evaluate the line integral over C of x ds for the following curve C. When C is a line segment joining (1, 1) to (4, 5). \int_Cxyz^2 ds , C is the line segment from (-1, 3, 0) to (1, 4, 1). Let u = (2, 2, 1) be a vector in R^3. F(x, y, z) = xye^zi + yze^xk, Find (a) the curl and (b) the divergence of the vector field. Evaluate the line integral over C of F*dr where F(x, y, z) = e^y i + xe^y j + (z + 1)e^x k and C is given by vector function r(t) = (t, t^4, 2t), 0 less than or equal to t less than or equal to 1. Evaluate the integral \int_0^1 {\left( {{e^{2t}}\vec i + {e^{ - t}}\vec j + t\vec k} \right)dt}. Consider the vector field F(x,y,z) = \langle -3yz, 6xz, 7xy \rangle Find the divergence and curl of F. a. Let F(x, y, z) = (x2 + y2)i + 2xyj + xyzk, (a) Find curl F. (b) Find div F. Let C be the curve represented by r(t) = ti + t^2 j, for t between 0 and 1. As a partial check, |v(1)| = \sqrt{30}, Evaluate the line integral c F d r , where C is given by the vector function r ( t ) F ( x , y , z ) = x i + y j + x y k , r t ) = cos t i + sin t j + t k , 0 t. Evaluate the line integral, where C is the given curve. \int_C xyz^2 ds, C is the line segment from (-2, 3, 0) to (0, 4, 1). Let B have a magnitude of 10 m, if A \cdot B = 43.3 m^{2}, find the angle \theta_{AB} between the vectors and the components for all possible so... Let \vec r\left( t \right) = t\vec i + {1 \over 2}{t^2}\vec j + {1 \over 3}{t^3}\vec k, find the unit tangent vector \vec T and the curvature k of \vec r(t) at the point t = 1. Evaluate \int_C xy \ ds where C is the line segment from (0,0) to (4,5). Vector Calculus: Understanding. Evaluate the line integral, where C is the given curve. If so, find a potential function for \vec F. Let (a, b) be a point in the plane, L a line y = mx + c. Find the point P on the line such that P minimizes the distance of the line to (a, b). Use the vectors U = 2i - 3j + 5k and V = 3i + 4j - 4k to find the expression. dr, where C is given by the vector function r(t). Find the equation of the tangent line to curve: x=3t-10; y=3t^{2/3} -12 at the point where t=8. Go ahead and submit it to our experts to be answered. The density of the wire at the point (x, y) is given by f(x, y) = 2x... Find the principal unit normal vector to the curve at the specified value of the parameter. Candidates can download Vector Calculus Study Materials along with Previous Year Questions PDF from below mentioned links.. Vector Calculus PDF Download. Exercises 174 22.3. F(x, y, z) = x^2yz i + xy^2z j + xyz^2 k, Find (a) the curl and (b) the divergence of the vector field. Evaluate I = \int_{C} (\sin x + 3y) dx + (8x + y) dy for the nonclosed path ABCD in the figure. \int_C x^2 dx - xy dy + dz, where c is the parabola z = x^2, y = 0 from (-1, 0, 1) to (1, 0, 1). True b. F(x, y) = xyi + 6y^2j r(t) = 17t^2i + t^3 j, 0 less than equal to t less than equal to 1. Evaluate the line integral, where C is the given curve. MATH 20550: Calculus III Practice Exam 1 Multiple Choice Problems 1. integral_C x y z^2 ds, C is the line segment from (-1, 6, 0) to (1, 7, 4), Find (a) the curl and (b) the divergence of the vector field. A vector v has initial point (0,0) and terminal point (2,3) . Find an equation of the tangent plane to the given parametric surface at the specified point. In this we have given Vector Calculus Study Materials for all competitive Exams like UPSC, MPPSC, APPSC, APSC, TNPSC, TSPSC etc. (b) In what directions starting at (0;ˇ=2) is fchanging at 50% of its maximum rate? Evaluate the line integral along the curve C. \int_C(xz + y^2) ds, C is the curve r(t) = (-5 - t)i + 2tj - 2tk, 0 \le t \le 1. U = 5i - 10j + 2k V = 10i + 20j - 6k. F(x, y, z) = xye^x i + yze^x k, Find the curl and the divergence of the vector field. The figure shows a section of a long, thin-walled metal tube of radius R = 4.52 cm, with a charge per unit length of 5.88 x 10^-8 C/m. The box shown in the image has a uniform magnetic field over each face. Find the equation of the plane through the point (4, -3, 3) and with normal vector 2{\bf{j}} - 3{\bf{k}}. Verify that the unit vector in the direction of v is ( 2/3, 2/3, +1/3). b. Decompose v into two vectors: v_1 and v_2 , where v_1 is parallel to w and v_2 is orthogonal to w . If vector{F} is a vector field in 3-dimension space, then grad(div vector(F)) = vector{0}. (3x^2 + 2xy^3)\vec{i} + (3x^2y^2 - 4y^3)\vec{j}, For the following vector field F, decide whether it is conservation or not by computing curl F. F(x, y) = -2yi-1xj, Find the magnitude of the resultant force and the angle it makes with the positive x-axis. Evaluate the line integral \int 2xy \ dx +3x^2 dy , where C consist of the line segment y = x^2 from (1, 1) to (2,4). Vector Calculus. Find th... Find the potential function f, such that gradient f = F = <2xy-y^3, x^2-3y^2x, sin z>. 1) to (1, 2). Evaluate integer_C xy^3 ds, where C is the first quadrant of the circle x^2 + y^2 = 9. Evaluate the line integral \int_Cx^5z ds, where C is the line segment from (0, 2, 7) to (6, 3, 8). Determine whether or not F is a conservative vector field. F(x, y, z) = xye^z i + yze^x k, Find (a) the curl and (b) the divergence of the vector field. Evaluate the line integral, where C is the given curve. F(x, y, z) = 1/square root{x^2 + y^2 + z^2} (xi + yj + zk), Find the curl and the divergence of the vector field. Consider the vector field F = \langle y\cos(xy),x\cos(xy), -\sin(z)\rangle and the curve C parametrised by r(t) = (\cos(t),\sin(t), 3 + \sin(t)) between the points (0, -1, 2) and (0, 1, 4). Compute the gradient vector field for the function, f(x,\ y) = 2x + 1y. F(x, y) = xy i + 3y^2 j r(t) = 6t^2 i + t^2 j, 0 \le t \le 1, Compute the curl of the following vector field. What is the magnitude of V_x? Evaluate the line integral integral_C F . Static GK topics for Competitive Exams – Check Static GK Competitive Exams || Download Study Materials Here!!!! Compute integral_C y dx + xy dy, if C goes from (0, 0) to (1, 3) along the line parameterized as x = t, y = 3t. Calculate Integral_{C} xy dx + (x+y) dy, where C is the path from (-1,1) to (2,4) along y=x^2. Let \vec{F}(x,y,z) = (x^3 \ln z)i + xe^{-y}j + (-y^2 - 2z)k Calculate curl \vec{ F } at P(1,1,1). x(t) = 3 sin t, y(t) = 3cos t; -π/2 \le t \le π/2. F(x, y, z) = x i + y j + xy k, \\ r(t)= \cos t i + \sin t j + t k, 0 \leq t \leq \pi. Compute the line integral of the vector field F=<4y,-3x>, over the circle x^{2}+y^{2}=25 oriented clockwise. Evaluate the line integral \int_C ydx -xdy, parabola y = x^2 for 0 \leq x \leq 5. Use Green's Theorem to evaluate the \int_{C}y^{3}dx-x^{3}dy where C is the circle x^{2}+y^{2}=9. Consider the vector field F(x, y, z) = (5yz, 4xz, 3xy) Find the divergence and curl of F. Decompose the vector v = 3i + 2j into two vectors v_1 and v_2, where v_1 is parallel to w = 2i + j, and v_2 is orthogonal to w = 2i + j. What is the overall magnitude of the force you're applying? Let C be the line segment from ( 0 , 0 , 0 ) to ( 1 , 2 , 3 ) . Apply Green's Theorem to evaluate the integral. F(x, y, z) = x^(3)yzi - x^(4)yK. r ( t ) = 3 cos ( t ) i + sin ( t ) j. F(x, y) = 1 / y i - y / {x^2} j. Previous Year Questions PDF Download Vector A has a magnitude of 15 and is pointing at an angle \theta_A = 62^\circ East of North. The curl of F = [{Blank}]. Find the length of the curve over the given interval. True or false? F(x,y)=\big, Find curl F for the vector field at the point (9,-9, 9). Find symmetric equations of the line normal to the surface z = f(x,\ y) = 3xy at (-2,\ -4,\ 24). Given the vectors, u and v, find the cross product and determine whether it is orthogonal to both u and v. u = <0, 7, 2>, v = <0, 0, -2>. Given P(2, 4) and Q(-3, -6), find the numerical form of the vector \vec{\mathbf {PQ}}. You are given vectors vector A = 5.0 hat i - 6.5 hat j and vector B = 3.5 hat i - 7.0 hat j. x = \boxed{\space}, \\y = \boxed{\... Two perpendicular forces are given by A = 6i - 4j + 2k and B = 3i - bj - k. Determine b. Evaluate the line integral \int_C {{\bf{F}} \cdot d{\bf{r}}}. Let C be the curve which is the union of two line segments, the first going from (0, 0) to (1, 2) and the second going from (1, 2) to (2, 0). Find the work done by the force field F in moving an object from P to Q F(x, y) = (2x/y, x^2/y^2); P(-3, 2), Q (1, 4). Determine whether or not the vector field is conservative. Evaluate the expression ||12i + j - 3k||^2. (a) Velocity of the particle for the specified vale of t. (b) Acceleration of the particle for the specified value of t. (c) Speed... Find the divergence of F(x, y, z) = x y i + (y^2 - 2 z) j + sin yz k, evaluated at the point (5, 2, pi/3).