(e) Verify Green's theorem for a vector field F(x, y)= y²i + x²j and a triangle bounded by the lines a+ y = 1 and -x + y = 1 and y = 0. (Hint: all theconditions in the Green theorem must beverified.)

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Answered: (e) Verify Green's theorem for a vector…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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(c) Let F(x, y, z) = x³i+y³j + z³k. Compute the following: (i) V-F (ii) V x F XB) (d) Give one example to illustrate that vectors satisfy the triangle inequality Justify your answer. Question Two (a) Prove that the vector field F(x, y, z) — (x² + yz)i – 2y(x + 3)j + (ry + 2)k is incompressible, and find its vector potential function.
1. Show that V × VG = 0 for all scalar field G(x,y,z).
The vector field F is shown below in the xy-plane and looks the same in all other horizontal planes. (In other words, F is independent of z and its z-component is 0.) (1) Is divF positive, negative, or zero? Explain. (2) Determine whether curl F = 0. If not, in which direction does curlF point?
Show that Green's Theorem holds for the vector fieldF(x, y)=(x +y, 3x) at a square of side 2.
it says "Use symbolic notation and fractions where needed" and this is my only attempy
Advanced Math Question
Definition: Let V be a vector space and U₁, U₂, ‚U be subspaces of V. Then V is said to be a direct sum of subspaces U₁, U2, ..., Uk, denoted by, V = U₁ ÐU₂ Ð···ÐUk, if the following two conditions hold: (i) V=U₁+U₂ +· ... + Uk; (ii) For every v € V, there exist unique vectors u¿ € Uį, 1 ≤ i ≤ k, such that V = U₁ + ... + uk. 9. (a) Suppose that U₁,...,Uk are subspaces of V. Prove that V = U₁ ... Uk if and only if the following two conditions hold: (i) V = U₁ + ... + Uk. Proved. (ii) The only way to write Oy as a sum of u₁ + … + Uk, where each u; € Uj, is by taking all u, i's equal to zero. Proved. (b) Suppose that V is a finite dimensional vector space, with dim(V) = n. Prove that there exist 1-dimensional subspaces U₁,...,Un of V such that V = U₁ U₂ Un. Proved. (c) Give an example to show that condition (ii) (in definition) can not be replaced with U₂ nu; = = {Ov}, for i ‡ j.
4. Assume A is a constant vector field and R = xi + yi + zk. (a) Simplify the expression (R. VR) · Ã+ V ·V × V × ((Ã × R) × (R × Ã)). (b) Assuming A the position vector of a point on the unit sphere centered at the origin, evaluate the above expression at R = A.
Verify Green's theorem for a vector field F(x, y) = y²³i+z²j and a triangle bounded by the lines z + y = 1 and -z+y = 1 and y= = 0. (Hint: all the conditions in the Green theorem must be verified.)
Show that the vector field A (3x²z + y?)ax + 2xya, + x³a, is conservative.
Q5: a) Show that the vector field F = (x² - y² + x)i- (2xy + y)j, is irrational.
(a) For all vectors ù, v, w € R³, ủ x ( xử) = ( xử) xử. True False (b) Let F: R³ True False R³ be a vector field of class C². Then div(curl F) = V. (V x F) = 0.

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(e) Verify Green's theorem for a vector field F(x, y) = y²i + x²j and a triangle bounded by the lines a +y=1 and -x + y = 1 and y = 0. (Hint: all the conditions in the Green theorem must be verified.)