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 *integral( [(bz – cy) dx + (cx – az) dy + (ay – ba) dz] ) equal to the plane area enclosed by 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 *integral( [(bz – cy) dx + (cx – az) dy + (ay – ba) dz] ) equal to the plane area enclosed by C".

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter9: Vectors In Two And Three Dimensions

Section9.6: Equations Of Lines And Planes

Problem 2E

See similar textbooks

Related questions

Q: 16. Let C be a simple closed smooth curve that lies in the plane x + y + z = 1. Show that the line…

Q: 8. Let S be the surface defined by the vector function R(u, v) = (v + 4, 4u - v, uv²). Set up an…

Q: Let C be a simple closed smooth curve in the plane 2x + 2y + z = 2, oriented as shown here. Show…

A: Given, The equation of the curve is 2x+2y+z=2. The closed integral is ∮C2y dx+3zdy-x dz Let…

Q: What is the normal vector, n for S is a surface of x +y² +z² = 36 in the first octant? %3D

Q: 3. Let C be the circle x + y = 16 and let S be any surface above the xy-plane with C as its…

Q: Find the surface integral F. dS of the vector field F(x, y, z) = (-x, -4, yz®) along the part of the…

Q: 1. In the plane, let D be a simply-connected region whose boundary OD is a piecewise C', simple,…

Q: Use the divergence (Gauss) theorem to find the value of (image 1) where F(x, y, z) = (4x3, ex2 − y,…

Q: (b) Let D be the disk of radius a in R², and let C denote it's boundary. Let N be the unit normal…

Q: a) Evaluate // (-xi- yj+2z k) · n dS, where o is the portion of the paraboloid z = 3x2 + 3y² between…

Q: Let S be the sphere x^(2)+y^(2)+z^(2)=4 with outward normal vector. Use the divergence theorem to…

Q: What is the value of F.ds, where F = 4xz,- y° i, + yz, ? Here, s is the surface bounded by x = 0, x…

A: Here function is given find f.ds as below

Q: 21. Show that the volume V of a region D in space enclosed by the oriented surface S with outward…

Q: Consider the surface S given by: 4x2 + 4z2 = (1+ y)?, Osys2.Compute: (a) a normal vector at each…

A: Given:- Consider the surface S given by: 4x 2 + 4z 2 = (1 + y) 2,0≤y≤2 Compute: (a)…

Q: Calculate the normal vector, n for the surface z=5-x-y with upward orientation.

A: We have to find the normal vector.

Q: Let F(x, y, z) = 2xi – 2yj + z²k. Evaluate Is where S is the cylinder x2 + y2 = 9 with z E [0, 1].…

Q: the normal line of the surface [z = (4x^2 + y^2] at the point (1,1,5) O 8x-2y+z=5 O 8x+2y+z=5…

A: We have to find the normal line of the surface.

Q: 4. Show that the line integral of F = [0, z] around a closed curve in the r- y plane, oriented as in…

Q: Let S be the disk enclosed by the curve C: r(t) = (cos cos t, sint, sin p cos t), for 0sts2r, where…

A: We need to find circulation.

Q: Given F(x, y, z) = x^2i + 2xyj + 3xzk and a region, being cut from the 1st octant by the sphere x^2…

Q: 4. Let F(r, y, z) =-ri+ yj + zk and o be the portion of the surface z = 8x-² - y? that lies above…

A: Given vector field is F(x,y,z)=43xi+43yj+53zk and also the surface σ is portion of the sphere…

Q: A normal vector to the tangent plane for the surface z = , at the point (1, 1, 1) is: y - Select…

Q: 6. Use the Normal Vector to a Smooth Parametric Surface to find the equation of the tangent plane to…

Q: b) Use Gauss theorem to evaluate F.ndS, where F(r, y, 2) = (2x²y² + In z) i – xy³ j – tanh(a² + y³)…

Q: b) Use Gauss theorem to evaluate F.ndS, where F(r, y, z) = (2x²y² + lIn 2) i – ry° j – tanh(x² + y³)…

A: Given function F(x, y, z) = (2x2y2+lnz)i -xy3j -tanh(x2+y2)k we have to find ∬σF.nds by using guass…

Q: Using Green’s theorem, evaluate the line integral ∮Cxydx+ (x+y)dy, where C is the curve bounding the…

A: Given: Where the components of the vector field are

Q: Find a vector normal to the surface x2 + y2 - z² = 18 at P = (3, 5, 4). %3D

Q: Evaluate F.N dS, where F(x, y, z) = 9x²i + 9xyj + 3x'y'k and S is the surface consisting of all…

Q: • Use Green theorem to evaluate the Line integral $(x-3y) da +(42+y) dy whe re C, the Yectungle with…

Q: Let F = 1xi + 5yj and let n be the outward unit normal vector to the positively oriented circle x² +…

A: To find the flux we have to first compute unit normal vectors by using formulan=Gradient…

Q: Let w = y dy A dz – x dz A dx + z dx A dy be a two-form on R°. Evaluate its surface integral I along…

A: w=ydydz-xdzdx+19zdxdy =yi^-xj^+z9k^i^dydz+j^dzdx+k^dxdy =∫∫yi^-xj^+z9k^ds→

Q: Let F(x, y,z) = (-xe² +3x, -ye² + arctan(x? + z), 2e²) and S the portion of the sphere acoording to…

A: Given that Fx,y,z=-xez+3x, -yez+arctanx2+z2, 2ez. Also, S=x,y,z| x2+y2+z2=1, z≥0. To determine the…

Q: 1. In the plane, let D be a simply-connected region whose boundary OD is a piecewise C, simple,…

Q: (b) Let D be the disk of radius a in R?, and let C denote it's boundary. Let N be the unit normal…

Q: Evaluate the line integral F. dr, where F(x, y, z) = 4xi + 4yj – 5zk and C is given by the vector…

Q: Evaluate an area of a surface where the vector that is normal to the surface is defined as; d. Q =…

Q: Use Stokes' Theorem to evaluate the line integral (y + 2z)dr + (x + 2z)dy + (x+ 2y)dz where C is the…

A: Given intgeral. ∫Cy+2zdx+x+2zdy+x+2ydz The Curve is formed by the intersection of the sphere…

Q: Find a unit normal vector to the surface f(x, y, z) = 0 at the point P(4, – 5, – 5) for the fur 5x…

Q: (a) Let S be the surface z2 = x² + y? bounded between the planes z = 0 and z = 2. Evaluate (z –…

Q: Use Gauss’ theorem to evaluate ∫∫σF.ndS, if σ is the closed surface of the hemisphere z=√25-x²-y²…

Q: 1. In the plane, let D be a simply-connected region whose boundary aD is a piecewise C', simple,…

Q: Find the flux of F over the closed surface. (Let N be the outward unit normal vector of the…

A: Given

Q: Compute the scalar line integral xy ds where C is the part of the circle of radius 3, centered at…

Q: Let F = yi + xj + x³y³ z³k, C the curve x2 + y² + 6x – 2y + 9 = 0, z = 2 (with counterclockwise…

Q: Let F(r, y) = 4xî + 3yî and let i be the outward unit normal vector to the positively oriented…

Q: Find the surface integral F. dS of the vector field F(x, y, 2) = (-æ, -y, yz®) along the part of the…

A: The complete solutions are given below

Q: Find a unit normal vector to the surface x² + 3y +z³ = 147 at the point (5, − 1, 5). O O O 1 √2,636…

Q: 2. Suppose that S is a surface in R3 with unit normal vector N surface area of S is given by (n1,…

Q: . Integrate f(x, y) = cos (x² + y²) (a) over the closed unit disk; (b) over the annular region 1 <…

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: 1. In the plane, let D be a simply-connected region whose boundary OD is a piecewise C1, simple,…

Question

With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.

Expert Solution