Please do (d) Green's Theorem(a) State the Green theorem in the plane.(b) Express part (a) in vector notation.(c) Give one example where the Green theorem fails, and explain how.(d) Prove part (a) assuming that the domain D enclosed by the simple closed curveC with positive orientation is of the formD = {(x,y) | a ≤ y ≤ b, gi(y) ≤ x ≤92(y)},where gi(y), 92(y) are continuous functions, and a, b are some real numbers.(e) Verify Green's theorem for a vector field F(x, y) = y²i+z²j and a trianglebounded by the lines a + y = 1 and -x + y = 1 and y = 0.(Hint: all the conditions in the Green theorem must be verified.)

Answered: Image /qna-images/answer/3ac146ae-61cb-4590-83ad-b4b264815e94.jpg

Answered: (d) Prove part (a) assuming that the…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.1: Length And Dot Product In R^n

Problem 83E: Guided Proof Prove that if u is orthogonal to v and w, then u is orthogonal to cv+dw for any scalars...

See similar textbooks

Similar questions

Guided Proof Prove that if u is orthogonal to v and w, then u is orthogonal to cv+dw for any scalars c and d. Getting Started: To prove that u is orthogonal to cv+dw, you need to show that the dot product of u and cv+dw is 0. i Rewrite the dot product of u and cv+dw as a linear combination of (uv) and (uw) using Properties 2 and 3 of Theorem 5.3. ii Use the fact that u is orthogonal to v and w, and the result of part i, to lead to the conclusion that u is orthogonal to cv+dw.
Properties of div and curl Prove the following properties of thedivergence and curl. Assume F and G are differentiable vectorfields and c is a real number.a. ∇ ⋅ (F + G) = ∇ ⋅ F + ∇ ⋅ Gb. ∇ x (F + G) = (∇ x F) + (∇ x G)c. ∇ ⋅ (cF) = c(∇ ⋅ F)d. ∇ x (cF) = c(∇ ⋅ F)
Flux of the radial field Consider the radial vector field F = ⟨ƒ, g, h⟩ = ⟨x, y, z⟩. Is the upward flux of the field greater across the hemisphere x2 + y2 + z2 = 1, for z ≥ 0, or across the paraboloid z = 1 - x2 - y2, for z ≥ 0?Note that the two surfaces have the same base in the xy-plane and the same high point (0, 0, 1). Use the explicit description for the hemisphere and a parametric description for the paraboloid.
A sphere is inscribed in a straight circular cylinder. The sphere is cut by any two planes perpendicular to the axis of the cylinder. Using Vector Calculus concepts, prove that the sphere and the cylinder part between the two planes have the same area. Use numerical or algebraic example.
Streamlines and equipotential lines Assume that on ℝ2, the vectorfield F = ⟨ƒ, g⟩ has a potential function φ such that ƒ = φxand g = φy, and it has a stream function ψ such that ƒ = ψy andg = -ψx. Show that the equipotential curves (level curves of φ)and the streamlines (level curves of ψ) are everywhere orthogonal.
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 = ⟨0, 0, -1⟩ across the slanted face of the tetrahedronz = 4 - x - y in the first octant; normal vectors point upward.
Maximum curl Let F = ⟨z, x, -y⟩.a. What is the scalar component of curl F in the direction of n = ⟨1, 0, 0⟩?b. What is the scalar component of curl F in the direction ofn = ⟨0, -1/√2, 1/√2⟩?c. In the direction of what unit vector n is the scalar componentof curl F a maximum?
Consider the scalar field (image provide). Use gradφ (image provide) to find a unit vector in the direction of gradφ. State how the direction of gradφ relates to surfaces of constant φ(x, y, z).
The mass in a mass-spring system (see figure) is pulled downward and then released, causing the system to oscillate according to x(t) = a1 sin ωt + a2 cos ωt where x is the displacement at time t, a1 and a2 are arbitrary constants, and ω is a fixed constant. Show that the set of all functions x(t) is a vector space.
(c) Sketch a condition to prove that the divergence of a curl of a vector is zero (0). Briefly explain your answer.
Using basis and dimensions in vector space section for linear algebra
General Advanced Linear Algebra question: What is an infinite-dimensional vector space, the definition, theorem, and an example of infinite-dimensional vector space?

SEE MORE QUESTIONS

Recommended textbooks for you

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Trigonometry (MindTap Course List)

Trigonometry

ISBN:

9781337278461

Author:

Ron Larson

Publisher:

Cengage Learning

Elementary Linear Algebra (MindTap Course List)

Algebra

ISBN:

9781305658004

Author:

Ron Larson

Publisher:

Cengage Learning

Algebra & Trigonometry with Analytic Geometry

Algebra

ISBN:

9781133382119

Author:

Swokowski

Publisher:

Cengage

Trigonometry (MindTap Course List)

Trigonometry

ISBN:

9781337278461

Author:

Ron Larson

Publisher:

Cengage Learning

SEE MORE TEXTBOOKS

(d) Prove part (a) assuming that the domain D enclosed by the simple closed curve C with positive orientation is of the form D = {(x,y) | a ≤ y ≤ b, gi(y) ≤ x ≤ 92(y)}, where gi(y), 92(y) are continuous functions, and a, b are some real numbers.