Kindly solve the task step by step 43-48. Surface integrals of vector fields Find the flux of the follow-ing vector fields across the given surface with the specified orientation.You may use either an explicit or parametric description of the surface.(0,0,-1) across the slanted face of the tetrahedronin the first octant: normal vectors point upward.43. F

Answered: Image /qna-images/answer/fd1b3716-eccf-4bae-805d-747682a9df07.jpg

Answered: (0,0,-1) across the slanted face of the…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter8: Applications Of Trigonometry

Section8.3: Vectors

Problem 60E

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a. Outward flux and area Show that the outward flux of theposition vector field F = xi + yj across any closed curve towhich Green’s Theorem applies is twice the area of the regionenclosed by the curve.b. Let n be the outward unit normal vector to a closed curve towhich Green’s Theorem applies. Show that it is not possiblefor F = x i + y j to be orthogonal to n at every point of C.
Find the flux of the following vector field across the given surface with the specified orientation. Use either an explicit or a parametric description of the surface. F=<−y,x,1> across the cylinder y=6x2 for 0≤x≤4, 0≤z≤2; normal vectors point in the general direction of the positive y-axis.
Please solve this question in 30 Minutes Question 4 Let Y define the surface Y: z=(x^2-y^2)/2 (x^2+y^2 less than or equal to 1) The surface is oriented in a way so that the unitnormal N has a positive z-component. Decide the flow of the vectorfield by the surface Y which means that you have to calculate the integral
The vectorfield r^2 shown in (Image 1) Let C be a part of an parabel in the form of y = ax^2 from point A(0,0) to B(1,2) . The given integral is (Image 2) Task is: Find a paramterization of the curve with the use of t, and calculate the integral with the use of definition. Then show that the vectorfield F is conservative, find the potential fucntion and calculate it.
The flow lines (or streamlines) of a vector field are the paths followed by a particle whose velocity field is the given vector field. Thus the vectors in a vector field are tangent to the flow lines. (a) Use a sketch of the vector field F(x, y) = xi − yj to draw some flow lines. From your sketches, can you guess the equations of the flow lines? (b) If parametric equations of a flow line are x = x(t), y = y(t), explain why these functions satisfy the differential equations dx/dt = x and dy/dt = −y. (c) Solve the differential equations to find an equation of the flow line that passes through the point (x, y) = (−1, −1).
Conservative fields Use Stokes’ Theorem to find the circulationof the vector field F = ∇(10 - x2 + y2 + z2) around anysmooth closed curve C with counterclockwise orientation.
This is a two part problem. Compute the flux of the vector field F = 2x^2y^2(k) through the surface S, which is the cone (sqrt(x^2 + y^2) = z), with z between 0 and R, oriented downward. A. Parameterize the cone using cylindrical coordinates. B. Find the flux of F through S.
Set up the surface integral to evaluate the flux of the vector field F=<y,x,x+z>, where S is the surface of the cylinder x2+y2=1 and the planes z=-1 and z=2 oriented outward. Please include 3D sketches of the 3 surfaces, sketches of the domain for each surface, normals for each surface. No need to evaluate to integral, just need help setting it up.
calc 3 13.7 #6 Evaluate the surface integral ∫∫S F · dS for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation. F(x, y, z) = y i + x j + z2 kS is the helicoid (with upward orientation) with vector equation r(u, v) = ucos(v)i + usin(v)j + v k, 0 ≤ u ≤ 5, 0 ≤ v ≤ 2?.
Surface integrals of vector fields Find the flux of the following vector field across the given surface with the specified orientation. You may use either an explicit or a parametric description of the surface. F = ⟨ -y, x, 1⟩ across the cylinder y = x2, for 0 ≤ x ≤ 1,0 ≤ z ≤ 4; normal vectors point in the general direction of thepositive y-axis.
An Eulerian flow field is described in Cartesian coordinates by V = 4i+xzj+5y3tk. (a) Is it compressible? (b) Is it steady? (c) Is the flow one-, two- or three-dimensional? (d) Find the y-component of the acceleration. (e) Find the y-component of the pressure gradient if the fluid is inviscid and gravity can be neglected.
(a) Let C be the curve of intersection of the surfaces x^2 = 2y and 3z = xy. Find the length of C from the origin to the point (6, 18, 36). 4. (b) Let f be a continuous vector field which is parallel to the unit tangent vector at each point of a smooth curve C'. Show that f. dr L || f| ds . (c) Let C" be a simple closed piecewise-smooth curve that lies in a plane with unit normal vector n = (a, b, c). Show that the line integral 1/2 *[(bz – cy) dx + (cx – az) dy + (ay – ba) dz] equal to the plane area enclosed by C".

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(0,0,-1) across the slanted face of the tetrahedron in the first octant, normal vectors point upward.