Let G(u, v) = (5u + v, 3u + 3v) be a map from the uv-plane to the xy-plane. Calculate Jac(G).

Q: Find the flux of F¯ = (x − y)i + xj across the circle x 2 + y 2 = 4 in the xy-plane

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Q: Let F = Use Stokes' Theorem to evaluate F. dī, where C is the triangle with vertices (9,0,0),…

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Q: Let G(u, v) = (2u + v, 7u + 3v) be a map from the uv-plane to the xy-plane. Describe the image of…

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Q: Use Green's theorem to calculate the work done by force F on a particle that is moving…

A: This is a greens theorem problem

Q: Let L denote the intersection line between the planes 2x-y-3z=4 and 2x+y-z=2 A) Determine a…

A: Solution of the system of equation will give the equation of the line of intersection.

Q: Let f be a function defined in R2 by f(x,y) = 3x2 - 2x3 - 3y2 + 6xy.  Find the gradient vector in…

A: We have to solve given problem:

Q: Find the curl of the curve F = (x*y,y'z,z'x)

A: As there are two methods of finding curl you can use any of them as I have shown below to get your…

Q: Integrate f(x,y) = √(64 - x2) over the smaller sector cut from the disk x2 + y2 ≤ = 64 by the rays θ…

A: Concept:   The calculus helps in understanding the changes between values that are related by a…

Q: Find a piecewise smooth parametrization of the path C. (ti + tj 0 sts 1 r(t) = 1st< 2 y = VI (1, 1)…

A: C:y=√x

Q: Calculate F dS if F(x, y, z) = x²yei+ xyj – 2xye k, where S is the surface of the pox bounded by the…

A: Given-F(x,y,z)=x2yezi+xy2j-2xyezk To find- ∬SF.dS ,where S is the surface of the box bounded by the…

Q: Since we have sin(r g) sin(r) sin(y), the function sin(r) is a linear transformation.

A: To justify whether the statement is true or false. As we know that,…

Q: Evaluate ſ F . n dS if F = (x + y² ™)ī − (2x)j + (2yz)k and $ is the surface of the plane 2x + y +…

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Q: Let f(x,y, z) = xy – yz and v [2y, 2z, 3x]. Find Curl(fv).

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Q: Use Stoke's Theorem to evaluate $. F · dR, where F (x, y, z) = (z²,xy, xz) and C is the triangle in…

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Q: Consider the vector-valued function [f(x, y) xt – xy² [g(x, y) 2 Vx + y³x² [f(x, y) Which of the…

A: Find the Jacobian of the given matrix

Q: Find the gradient vector field of f(x, y, z) = x³y* + y° z³ + xz°.

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Q: In ANPQ, U is the centroid. If NS=15 find US. R S U

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Q: Integrate the function f(r,y, z) = Over the entire 3D coordinate system, i.e., all of R'.

A: We have to integrate the function fx, y, z=1x2+y2+z21+x2+y2+z2 over the entire 3D coordinate system,…

Q: = xyi + yzĵ + de is a in x-y plane.

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Q: Find the first and second fundamental form of x(u,v)=(e(u-v)/2 cos(u+v)/2, e(u-v)/2…

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Q: Find a piecewise smooth parametrization of the path C. 0 st<1 (ti+tj r(t) = %3D 1<t<2 (1, 1) y=x

A: Given rt=ti + tj                             0≤t≤1                                       1≤t≤2

Q: Sketch the surface z = f(x,y). f(x.y) = 4x² + 4y² + 2 OC. OD.

A: Given function is fx,y=4x2+4y2+2. We have to sketch the surface of fx,y=4x2+4y2+2.

Q: Let F1 = (x+ y)Âx + (-x+y) ŷ– 2zî and F2 = 2 yx+ (2x +3z) ŷ +3yî. Calculate the curl and divergence…

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Q: Find the circulation of F over the counterclockwise closed curve C: F= tan-1yi+ + 2x + y+z)j +yk…

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Q: Find the gradient vector field (F(r, Y, 2)) of f(x, y, z) = In(4x + 6y + 22) . F(7, y, 2) = (

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Q: Evaluate the surface z =1,0<x,y <1.

A: To solve this we will use divergence theorem.

Q: Find the null space of the linear transformation T: R →R TE, y, z) = (x + y, z). defined by

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Q: Let F Use Stokes' Theorem to evaluate F. dī, where C is the triangle with vertices (8,0,0),…

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Q: Evaluate / . F. dS where F = and the surface S is the triangle with vertices (1, 1, 0), (0, 2, 0),…

A: First, we plot the surface S, and try and form its equation.Since S lies in a plane, it is the part…

Q: Let (u, v) = (4u + v, 2u + 9v) be a map from the uv-plane to the ry-plane. Describe the image of the…

A: The given transformation Φ is, x=4u+v⋯⋯1y=2u+9v⋯⋯2 Let's express u and v in terms of x and y. In…

Q: Given the joint PDF of X and Y: f(x, y) = e-(*+y), x> 0, y> 0; 0 otherwise. Let U V = X + Y. Using…

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Q: a) Evaluate Fự ds, where the curve C is the intersection between the surface z = V1- y? and the…

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Q: Use the transformation x = * and y = uv (with u > 0 and v > 0) to evaluate /| xy dA R where Ris the…

A: The integral is ∬Rxy2dA. Transformation: x=uv and y=uv We obtain the integral as follows:  ∬Rfx,…

Q: Use the Jacobian of a coordinate transformation to find the area of the parallelogram bounded by the…

A: We use the following transformation to find the area of the parallelogram bounded by the lines y =…

Q: (a) For a scalar function f(r), show that the curl of the gradient is zero. That is, V (

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Q: Given F = xyi + 1(x – y) j+ 1(x + y) k, (1) Compute the fundamental cross product F, xF, =

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

Q: Let C be the intersection of the surface x² + y² + z² cylinder x² + 3y² = 4, z > 0. = 9 and the

A: As per the question we are given the equation of a sphere and a cylinder (with elliptical cross…

Q: find the flux of f=zk across the portion of the sphere x2+y2+z2=a2 in the first octant in the…

A: Given the field, F=zk The surface S is the portion of the sphere x2+y2+z2=a2 in the first octant in…

Q: Find the circulation of F over the counterclockwise closed curve C: F= tan y i+(, +x +z)j+yk 1 + y²…

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Q: Sketch the surface z = f(x,y). f(x.y) = 4x2 + 4y2 + 2 O A. OB. OC. OD. N

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Q: Let F(x, y) = (2x + y³, 2y – x4). Compute the flux oF. F.n ds of F across the circle of radius 5…

A: Given function can be written as, Region bounded by the circle of equation,

Q: Calculate dS where F = (e* sin y, e" cos y, yz²) and S is the surface of the box bounded by the…

A: We need to find the surface integral. We will need basic knowledge of parametrization of vector…

Q: Let F(r, y, z) = ye²i – rzj + cos zk. Compute curl F. %3D

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Q: Let r(t)=⟨sin(t),cos(t),9sin(t)+3cos(2t)⟩. Find the projection of r(t)r(t) onto the xz-plane for…

A: Set y=0 as the answer is asked for xz plane.

Q: Find a conformal map f (z2) which takes the disk {z : 2 < 3} onto the left half plane {w: Re(w) }and…

A: Given clear step by step explanation below

Q: Let F(x, y, z) = (x,y,z). Compute ∫∫sF · dS where S is the part of the plane x + y + z = 1 in the…

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Q: Let F = (r-y, z+y+3, z2) and S y-direction. Compute {(r, y, z) : y =1-r²-z², y20} oriented in the…

A: Here orientation is positive y direction, surface is open.  During the calculation we ignore many…

Q: Since we have sin(r +y) sin(r) + sin(y), the function sin(r) is a linear transformation.

A: We know T: V---&gt;W  is said to be linear transformation if, T(x+y)= T(x)+T(y) for all x,y in V…

Vertices Of An Ellipse

In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is the intersection point of the main axis. In an ellipse graph, the center of the axis is the midpoint of the line segment connecting the foci. The major axis is the line section that runs through the ellipse's foci, while the minor axis runs through the middle and is perpendicular to the major axis.. The ellipse has two endpoints on the major axis which are known as the vertices (in plural), vertex in the singular. The equation of the major axis is 2a, the equation of the minor is 2b and the distance between the major axis and minor axis is 2c. The semi-major axis has a length of a and the semi-minor axis has a length of b. The ellipse's vertices are determined by the elliptical orbit's equation and graph.