Proof of Stokes’ Theorem Confirm the following step in theproof of Stokes’ Theorem. If z = s(x, y) and ƒ, g, and h are functionsof x, y, and z, with M = ƒ + hzx and N = g + hzy, then My = ƒy + ƒzzy + hzxy + zx(hy + hzzy) and Nx = gx + gzzx + hzyx + zy(hx + hzzx).

Proof of Stokes’ Theorem Confirm the following step in theproof of Stokes’ Theorem. If z = s(x, y) and ƒ, g, and h are functionsof x, y, and z, with M = ƒ + hzx and N = g + hzy, then My = ƒy + ƒzzy + hzxy + zx(hy + hzzy) and Nx = gx + gzzx + hzyx + zy(hx + hzzx).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter8: Applications Of Trigonometry

Section8.3: Vectors

Problem 60E

See similar textbooks

Related questions

Q: A space curve Let w = x2e2y cos 3z. Find the value of dw/ dt at the point (1, ln 2, 0) on the curve…

Q: Heat flux The heat flow vector field for conducting objects is F = -k∇T, where T(x, y, z) is the…

Q: Conservative fields Use Stokes’ Theorem to find the circulationof the following vector field around…

Q: Scalar line integrals Evaluate the following line integral along the curve C.

Q: QI Given the electric flux density, D= xya, + xya, + xya, C/m2. By using divergent theorem find Q…

A: NOTE: We are entitled to solve only the first question, unless specified. Formula Used:Divergent…

Q: tate and Prove Euler's Theorem on homogeneous functions.

A: Statement : Euler's theorem states that if f is a homogeneous function of…

Q: Using Green's theorem find the value of f F.drWhere F(x,y) = (e* – y³)i + (cosy + x³)j and C is the…

Q: Determine whether the statement is true or false. Stokes' Theorem is a special case of Green's…

A: The given statement is Stokes' Theorem is a special case of Green's theorem. Stokes' Theorem is…

Q: Flux integrals Compute the outward flux of the following vector fields across the given surfaces S.…

Q: Calculate curl(F) and then apply Stokes' Theorem to compute the flux of curl(F) through the surface…

A: F→=x+y,z2-64,xy2+1F→.dr→=x+ydx+z2-64dy+xy2+1curl…

Q: pdf. Let f(z) =z. Find dz from z=0 to z=4+2i alongthe curve C given z=2i to z=21.

Q: 7. Use Stokes' Theorem to evaluate f, F· dr where C is oriented counter- clockwise as viewed from…

A: Given Fx,y,z=yi-zj+x2kC is the triangle with vertices 1,0,0, 0,1,0, 0,0,1and the triangle is a…

Q: Find div F and curl F F(x,y,z)=x^2i-2j+yzk F(x,y,z)=xz^3i+2y^4x^2j+5z^2yk

A: Given:- i) F(x,y,z)=x2i -2j+yzk ii )F(x,y,z)=xz3 i+2y4x2j +5z2 yk To find:- div F and curl F of…

Q: 7. Stokes' Theorem is a generalization of the (a) fundamental theorem of line integrals. (c)…

Q: a) Show that F (x, y) = (yexy + cos(x + y)) i + (xexy + cos(x + y) j is the gradient of some…

A: Given the function F(x,y) is the gradient of some function.

Q: outward flux of the vector field F = (x cos y, 2 sin y, 3z cos y) across the Decide which integral…

A: Given vector field is F→=xcosy,2siny,3zcosy and S is the boundary of the region bounded by the…

Q: Find real differentiable functions s(y) and t(y), such that the complex function f(x + iy) = e"…

Q: Use Stokes's Theorem to evaluate F. dr. In this case, Cis oriented counterclockwise as viewed from…

A: given F(x,y,z)=2yi+3zj+xk

Q: ntegral calculus A particle moves according to the given acceleration function over the given…

A: Disclaimer: Since you have asked multiple questions; we will solve the first question for you. If…

Q: Use Stokes's Theorem to evaluate F. dr. In this case, C is oriented counterclockwise as viewed from…

A: Using the Parametric equation of the plane.containing the triangle

Q: force F = 5cos3t. Determine the position of the mass as a function of time. Given the differential…

A: We want to find correct answer.

Q: Scalar line integrals Evaluate the following line integral along the curve C.

A: Parameterize: x = 4cosθ; y = 4sinθ x2 + y2 = r2 ds = rdθdr

Q: Green's Theorem (8x – f,) dA = f dx + g dy %3D (Circulation form)

Q: A normal line to a surface S at a point (x, y, z) E S is a line perpendicular to the tangent plane…

Q: se Stokes' Theorem to evaluate | F. dr where C is oriented counterclockwise as viewed from above.…

Q: Use Stokes's Theorem to evaluate JcF•dr. Use a computer algebra system to verify your results. C is…

Q: Verifying path independence Consider the potential functionφ(x, y) = (x2 - y2)/2 and its gradient…

Q: Use Stokes's Theorem to evaluate ·[F. Use a computer algebra system to verify your results. C is…

Q: Let F . %3D Use Stokes' Theorem to evaluate / curlF dS, where S consists of the top and the four…

A: Given the vector field F→=xyz,xy,x2yz. Let S be a cube whose diagonal is the line joining (2,2,2)…

Q: 10. Show that the function f(x,y, z) = z arctan satisfies the Laplace equation.

A: We have to cheek

Q: Dierentiable curves with zero torsion lie in planes That a sufficiently di¡erentiable curve with…

Q: Use Stokes's Theorem to evaluate Jc F. dr. Use a computer algebra system to verify your results. C…

A: Formula is in the solution part

Q: Use Stokes's Theorem to evaluate F. dr. In this case, C is oriented counterclockwise as viewed from…

A: we have, Fx,y,z=2yi^+3zj^+xk^ and vertices of triangle 3,0,0,0,3,0,0,0,3 ∇×F=ijk∂∂x∂∂y∂∂z2y3zx…

Q: Question: Using Cauchy Riemann equations prove that the followings do not have derivatives: (Assume…

A: According to Cauchy Riemann equation if fz=ux,y+ivx,y is analytic then ux=vy and uy=-vx. If a…

Q: region bounded by the straight lines 3x + 2y = 6, x = 0 and y = 0. Let F(x,y) = (ax?y +y³ + 1)i +…

Q: 5. Test the Stokes' theorem for the function: V 3xy'x-zcos (2x)ÿ + xy°ż .Do it for the plane surface…

Q: Using Stokes's Theorem In Exercises 83 and 84, use Stokes's Theorem to evaluate F• dr. In each case,…

Q: Evaluating line integrals Use the given potential function φ of the gradient field F and the curve C…

A: a) Calculate the value of gradient field ∇φ, Find ∇φ at r(t), Calculate r'(t),

Q: se the Divergence Theorem to evaluate S F · N dS and find the outward flux of F…

Q: Line integrals Use Green’s Theorem to evaluate the following line integral. Assume all curves are…

A: Green theorem can be represented as, Write the functions for P and Q, Region is bounded by a…

Q: Work along different paths Find the work done by F = (x2 + y)i + (y2 + x)j + ze*k over the following…

A: a.

Q: (2,2) (0,2) II (0,0) (2,0) Exercise 1 Let of = (2xy – x'y) dx + (x² – x²y) dy (a) Show that df is an…

A: Differential equation

Q: a particle along line segments from the origin to the points (2, 0, 0), (2, 5, 3), (0, 5, 3), and…

Q: Gradient fields on curves For the potential function φ and points A, B, C, and D on the level curve…

A: Given function is: φ(x, y) = y - 2x The Gradient of φ(x, y) is given by: F = ∇φ

Q: Use Stokes's Theorem to evaluate F. dr. In this case, C is oriented counterclockwise as viewed from…

Q: Surface integrals: Calculating the flux over a particular surface is a very common way of determine…

A: According to our company policy, I am supposed to answer only one question at a time. For the rest…

Q: Surface integrals of vector fields Find the flux of the following vector field across the given…

Q: derivatives of u, v and w with respect to x, y and z, and use them to compute Jacobian Obtain the…

Q: A force field F moves a particle with mass m > 0 along a trajectory C in space parametrized by r(t)…

Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

Proof of Stokes’ Theorem Confirm the following step in the
proof of Stokes’ Theorem. If z = s(x, y) and ƒ, g, and h are functions
of x, y, and z, with M = ƒ + hz_x and N = g + hz_y, then
M_y = ƒ_y + ƒ_zz_y + hz_xy + z_x(h_y + h_zz_y) and
N_x = g_x + g_zz_x + hz_yx + z_y(h_x + h_zz_x).

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Differential Equation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Use Stokes's Theorem to evaluate C F · dr. C is oriented counterclockwise as viewed from above. F(x, y, z) = (cos(y) + y cos(x))i + (sin(x) − x sin(y))j + xyzk S: portion of z = y2 over the square in the xy-plane with vertices (0, 0), (a, 0), (a, a), and (0, a)
Use Green's Theorem in the form of this equation to prove Green's first identity, where D and C satisfy the hypothesis of Green's Theorem and the appropriate partial derivatives of f and g exist and are continuous. (The quantity ∇g · n = Dng occurs in the line integral. This is the directional derivative in the direction of the normal vector n and is called the normal derivative of g.)
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.
Conservative fields Use Stokes’ Theorem to find the circulationof the following vector field around any simple closed smooth curve C. F = ⟨2x, -2y, 2z⟩
Gradient fields on curves For the potential function φ and points A, B, C, and D on the level curve φ(x, y) = 0, complete the following steps.a. Find the gradient field F = ∇φ.b. Evaluate F at the points A, B, C, and D.c. Plot the level curve φ(x, y) = 0 and the vectors F at the points A, B, C, and D. φ(x, y) = y - 2x; A(-1, -2), B(0, 0), C(1, 2), and D(2, 4)
Double integral to line integral Use the flux form of Green’sTheorem to evaluate ∫∫R (2xy + 4y3) dA, where R is the trianglewith vertices (0, 0), (1, 0), and (0, 1).
Use Stokes's Theorem to evaluate F · dr C . In this case, C is oriented counterclockwise as viewed from above. F(x, y, z) = 2yi + 3zj + xk C: triangle with vertices (5, 0, 0), (0, 5, 0), (0, 0, 5)
Stokes’ Theorem on closed surfaces Prove that if F satisfies theconditions of Stokes’ Theorem, then ∫∫S (∇ x F) ⋅ n dS = 0,where S is a smooth surface that encloses a region.
Heat flux The heat flow vector field for conducting objects is F = -k∇T, where T(x, y, z) is the temperature in the object and k > 0 is a constant that depends on the material. Compute the outward flux of F across the following surfaces S for the given temperature distributions. Assume k = 1. T(x, y, z) = -ln (x2 + y2 + z2); S is the sphere x2 + y2 + z2 = a2.
A space curve Let w = x2e2y cos 3z. Find the value of dw/ dt at the point (1, ln 2, 0) on the curve x = cos t, y = ln (t + 2), z = t.
Work in a hyperbolic field Consider the hyperbolic force field F = ⟨y, x⟩ (the streamlines are hyperbolas) and the three paths shown in the figure. Compute the work done in the presence of F on each of the three paths. Does it appear that the line integral ∫C F ⋅ T ds is independent of the path, where C is any path from (1, 0) to (0, 1)?
StokesTheorem.Evaluate∫ F·dr,whereF=arctanx/yi+ln√x2+y2j+k and C is the boundary of the triangle with vertices (0, 0, 0), (1, 1, 1), and (0, 0, 2).