Verify Green's Theorem by evaluating both integrals [ 1² dx + x² oy = √ √ (2x - 3) da dA dy əx JR for the given path. C: square with vertices (0, 0), (8, 0), (8, 8), (0, 8) [x² dx + x² dy = [ le/ (x - 2) = [ dA dA əx

Q: 2. If F = (3x² - 3yz)i + (3y²-3xz)-(32²-3xy find div curl

A: Given problem:-

Q: Let P denote the path that travels along the graph of y = x2 from (0,0) to (1,1). Let f(x,y) = √1 +…

A: Let P denote the path that travels along with the graph of y=x2 from (0, 0) to (1, 1). Let f(x,…

Q: The figure below shows the expression for the x-centroid of line L. Suppose length L is 10 cm and…

A: Suppose L is the length of the line. Then, the x-coordinate of the centroid of the line is…

Q: Calculate the line integral f -y²dx + xdy where ŕ is the path y² = 2x − x² such that x>1 and y > 0.

A:

Q: 3. Compute the inner product g (v,w) of v = + x. 2 əx and w = x.-+ 2y. Əx ay' Where the line element…

A:

Q: Given A = (a₁, a2, a3) and B = (b₁,b2, b3) a parametrization of the segment line joining B to A is…

A: Given points, A=a1,a2,a3  and  B=b1,b2,b3

Q: 1 3 - -- () ---- () 5 V2 = 10 3 12 4. Let VI: a) For what values of his v3 in Span {V₁, V2} ? b) For…

A: The given vectors,.The objective is to find the values of  for which the vector  is in the  and  are…

Q: If u xv=uxw and u ‡ ♂, then u = w.

A: The given statement is: If u×v=u×w and u≠0→, then u=w

Q: Let C be the positively oriented square with vertices (0, 0), (2, 0), (2, 2), (0, 2). Use Green's…

A: Introduction: According to Green's Theorem, a line integral of a vector over a closed curve C…

Q: 5. Evaluate the integra 21-dA, RZ-3y %3D where R is the parallelogram enclosed by the lines 2r-y 0,…

A:

Q: 4. Let C be the unit circle in the xy-plane, and S be the boundary of the square with side 1,…

A:

Q: 3. Evaluate the line integral (3ry² + 6y) da, where C' is the path traced by first moving from the…

A: The given problem is to evaluate the given line integral of (3xy2+6y) over the given path C which is…

Q: 12 Consider the DE: x' x, with solutions x = est [3e L4e-2t and x, = a) Verify that x, and x2 are…

A: By finding the wronskian of the solutions

Q: Suppose F (x, y) = (2xe-", 2y – a²e="} (a) Find a scalar function f (x, y) such that F = Vf (b) Use…

A:

Q: Let C be the positively oriented square with vertices (0, 0), (2,0), (2, 2), (0, 2). Use Green's…

A: Let the given square be OABC which is positively oriented.

Q: ented square v V , (0, 1). The va + 8x² y dy)

A:

Q: Suppose that we start at the origin in the xy-plane and take n unit steps (i.e., each step is of…

A:

Q: e the Cauchy problem y' = y cosx, y(0) = 1.

A:

Q: f (lnx + y) dx − a² dy, where & is the rectangle with vertices (1, 1), 1. Use Green's Theorem to…

A: Let I =∮(ln x+y)dx-x2 dy, where the region is C which is a rectangle with vertices…

Q: where Dx = X₁-X₂ and Dy = |Y₁ — Y2]. Assume that the two locations are independent and uniformly…

A:

Q: Let C₁ be a straight line in the Argand plane from (0, 0) to (3, 3), and let C₂ be the path…

A: We have to calculate the following integrals i∫C1z2dz  and ii∫C2z2dz where C1 and C2 are given…

Q: If F(x, y, z) = (sin x + 2) i+2xz j+ (e-x) k and the path C is the line segment from A(1,2, 1) to…

A:

Q: ndwritten please Rew

A: Introduction: A Boundary price drawback may be a system of normal differential equations with…

Q: Prove the following integration by parts statement: Let f and g be holomorphic in G, and suppose y C…

A:

Q: 6. Show that the line integral fry²dx + (x²y + 2x)dy around an rectangle depends only on the area of…

A: Answer :-

Q: (a) Use a Green theorem to evaluate (y¹ - 2y)dx - (6x - 4xy³). dy where c is the boundary of the…

A:

Q: Compute ,f for f(2) +6 + 6i, positively oriented. sin(32) and path 2 that is the boundary of the…

A: Let's see the method of integration. We use residue method to solve this integral.

Q: Two points in rectangular coordinates are given by P₁ (0,0,2) and P₂ (0,1, √3). Obtain the line…

A: We are given points in rectangular coordinates are given by P1(0, 0, 2) and P20, 1, 3. Here, we need…

Q: x ds, where C is the portion of the parabola from (0,0) to (1, 1), followed by the line segment from…

A:

Q: Let S(x) be a natural cubic spline that passes thorugh the points (-1,1), (0,2), and (1,1). Then,…

A:

Q: 6. Evaluate S (x² + y²) ³/2 dydx where D is the disc x² + y² = 4. Show your work!

A: