# vector calculus questions and answers pdf

## vector calculus questions and answers pdf

(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. 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. 0 is the initial position vector and !v 0 is the constant velocity vector of the object. Calculate the curl (rotation) of V . Previous Year Questions PDF … Represent the plane curve \dfrac{x^2}{16} -\dfrac{y^2}{4} = 1 by a vector-valued function. Determine if the vector field \vec F = -\frac{y}{x^2+y^2} \hat i + \frac{x}{x^2+y^2} \hat j is conservative or not. True or false? Let v = -7i + 3j , and w = -i - 6j . Verify that the unit vector in the direction of v is ( 2/3, 2/3, +1/3). F(x, y, z) = xye^x i + yze^x k, Find the curl and the divergence of the vector field. Compute fox \int_Cxds where C is the parabola y = x^2, with 0 \le x \le 1. Evaluate the line integral \int_{C} 4ry^6 ds, where C is the right half of the circle x^2 + y^2 = 36 \\\boxed{\space}. {\bf{F}} = {{\left\langle {x,y,z} \right\rangle } \over {5 + 4{x^2} + 7{y^2}}}, Find the divergence of the following vector field. Let u = 6\hat i \times 7\hat j \text{ and } v = \hat j \times \hat k. Compute u \times v \text{ and } v \times u. dr, where F(x, y, z) = 4 sin x i + 2 cos y j + 5xz k and C is given by the vector function r(t) = t^3 i - t^2 j + t^1 k, 0 less than or equal to t less tha... Two forces act on an object. Find the curl and the divergence of the vector field. Find integral along C 3(x-y) ds. Given R(t) = e^{4t} \cos(t) \mathbf i + e^{4t} \sin(t) \mathbf j + e^{4t} \mathbf k. Find the derivative R'(t) and norm of the derivative. To give you a feeling for the issues, suppose you were interested in the temperature T of water in a river. Vector B has a magnitude of 24 and is pointing at an angle \theta_B = 72^\circ south of west. integral_C 6y^3 dx - 6x^3 dy C is the circle x^2 + y^2 = 4. (a) Parameterize C by a curve r(t), 0 less than equal to t less than equal to 1. Evaluate C ( 2 x y ) d x + ( x + 2 y ) d y . The vector V_3 in the diagram is equal to: A.\ V_1 - V_2\\ B.\ V_1 + V_2\\ C.\ V_2 - V_1\\ D.\ V_1\cos\theta\\ E.\ \dfrac{V_1}{\cos\theta}. Suppose r(t) = t i + t^2j + 5tk. A. False. Find the magnitude and direction of the plane's flight path. F(x, y) = xyi + 6y^2j r(t) = 17t^2i + t^3 j, 0 less than equal to t less than equal to 1. F(x, y, z) = xye^zi + yze^xk, Find (a) the curl and (b) the divergence of the vector field. 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). Get help with your Vector calculus homework. 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. Using the parameterization method, evaluate integral_C vector F . Use Green's Theorem to evaluate the following line integral. Use Green's Theorem to compute the integral \int_C (x^2y+x)dy +y^2xdx where C is the semi-circle x^2+y^2 = 209 and y \geq 0. Find the principal unit normal vector N(t) at the point (5\pi, 1,0) for the curve defined by r(t) = 5t\, i + \cos(12t)j+\sin(12t)k. _____ A) \left ( -\frac{144}{13} \right )\cos\left ( 12t \right )... Find the curvature, K, of the plane curve at the given point. Candidates can download Vector Calculus Study Materials along with Previous Year Questions PDF from below mentioned links. Use Green's Theorem to find the counterclockwise circulation and outward flux for the field F = (8x-7y)i + (4y-7x)j and curve C: the square bounded by x= 0, x=7. A vector v has initial point (0,0) and terminal point (2,3) . Determine if the vector field F =\langle y z, x z, x y \rangle is conservative. Apply Green's Theorem to evaluate the following integral: oint C (3y dx + 2x dy), C: the boundary of 0 leq x leq pi, 0 leq y leq sin x. Earn Transferable Credit & Get your Degree. Compute the line integral \int_{C}ydx+x^{2}ydy where C is the boundary of the square 0\leq x\leq3, oriented in the counterclockwise direction. Find the curve integral of (3x - 2y) ds. 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 . Calculus and Vectors. Find div F. \int_C( x^2+y^2)ds \ \ \ C: the line segment from (-1,-1) to (2, 2) . Evaluate the line integral \int_Cydx + xdy where C is the parameterized path x = t^2, y = t^ 3, 1 \leq t \leq 6. VECTOR CALCULUS I YEAR B.Tech . Label the mean and the inflection points. Evaluate int (x+y) ds over C, a line segment from (0,2,0) to (2,0,0). Evaluate C x y z d s. Find a vector of magnitude 7 units in the direction of the vector . Evaluate the expression ||12i + j - 3k||^2. Write v in the form v = ai + bj. curl F = _____ (b) Find the divergence of the vector field. F(x, y, z) = 9e^(xy) sin(z)j + 8y tan^(-1)(x/z)k. A) Find the curl of the vector field. Suppose a = 4i - 3j, b = \left \langle -2, 1 \right \rangle,\ and\ c = \left \langle 1, -2 \right \rangle Write vector a in trigonometric form. Draw a normal curve with mu = 57 and sigma = 16. Calculate the integrals either directly or using Fundamental Theorem of Calculus: C y z d x + 2 x z d y + e x y d z where C is the circle x 2 + y 2 = 16 , z = 5 . 0 B. d\vec{r} for \vec{F} = - 4y\vec{i} - x\vec{j} - 4z\vec{k}. Vector B has a magnitude of 24 and is pointing at an angle \theta_B = 72^\circ South of West. Use Green's Theorem to evaluate \int_C F\cdot dr. Find div F. Compute \int_C y dx + xy dy if C goes from (0, 0) to (1, 3) along the line parameterized as x = t, y = 3t. x = \boxed{\space}, \\y = \boxed{\... Two perpendicular forces are given by A = 6i - 4j + 2k and B = 3i - bj - k. Determine b. Vector Calculus Solutions to Sample Final Examination #1 1. _____ (smaller i-value) _____ (larger i-value). Let C be the positively oriented square with vertices (0, 0), (3, 0), (3,3), (0, 3). Calculate int C xy + 2 dx where C is the line segment from (1, 1) to (2, -1). Determine if the vector field F = (-yx^2 + y^2)\hat i + (xy^2+y^2)\hat j is conservative of not. Determine if this vector field is conservative. The problems are sorted by topic and … Apply Green's Theorem to evaluate the integral. Evaluate the line integral Integral_C (x^2 + y^2) dx + 2xy dy, where C is the path of the semicircular arc of the circle x^2 + y^2 = 25 starting at (5, 0) and ending at (-5, 0) going counterclockwi... A very long straight wire has charge per unit length 1.57 times 10^{-10} C / m. At what distance from the wire is the magnitude of the electric field equal to 2.56 N / C? Let F = (2xy + z^3) i + (x^2) j + (3xz^2) k. Find the work done to move a body in this field from (1, -2, 1) to (3, 1, 4). Let F(x, y, z)=(cos y + ycos x)i+(sin x-xsin y)j+yzk. Find r(t), if r'(t) = 4sin(2t) i + 8t3 j and r(0) = 8 i - 3j. Find the vector components of r. Show your work and provide a diagram.

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