Problem 4. (Flux) Given the surface M defined by the parametric formula r(u, v) = (u, v, u² + v²) where 0≤u≤ 1,0 ≤ v ≤ 1 and the vector field compute the flux JJM² F.dS F = (P, Q, R) = (-z, xy,x) =ff Pdy ^dz + Qdz A dx + Rdx / M A dy

Q: 3. Let C be the circle x + y = 16 and let S be any surface above the xy-plane with C as its…

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Q: 3. Find the directional derivative of the given function f at the given point P in the direction of…

A: 3. From the given problem:   fx,y,z=x2+y2+z2 ; p1,2,2 ; a=3,2,1   As we know that the expression for…

Q: 49. Let f(x, y, z)=sin(x² + y² +2²). (a) Describe in words the shape of the level surfaces of f. (b)…

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Q: 2. Use Green's Theorem to find the flux of F(x, y) = (-y², xy) across the square with vertices…

A: Green's Theorem:The circulation of vector field F=<M,N>in a closed path and the flux of the…

Q: 5. Let F(x, y) = square region with vertices (0,0), (1,0), (1, 1), and (0, 1) oriented…

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Q: Suppose that r(u)r(u) is a vector valued function of uu, and suppose…

A: As per the question we are given a Parametrized curve r(u) in 3d and its derivative at u = 3 as:…

Q: Given the 2-D vector field : G(x, y) = (2y)î + (x)j and the triangular curve T : line segments from…

A: Given: G→x,y=2yi^+xj^ Triangular curve T, line segments from point A0,0 to point B4,0 to point D4,2…

Q: 6. Consider the system x' = 2r y=-y. I (a) Sketch the vector field. (b) Show that for every solution…

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Q: The normal line to the surface x² + 2y² + ² = 25 at the point (1, -2, 4) is parallel to the vector O…

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Q: 1. Given f(x,y) = 2x² + y² whose graph is a paraboloid. Fill in the table with the values of the…

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Q: Find an expression for a unit vector normal, no, to the surface x = (4 – cos (v)) cos (u), y = (4 –…

A: The complete solutions are given below

Q: 6. Let f(x, y) = ln(2x - y²). (a) (b) (c) Determine the gradient of f at the point (2, -1).…

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Q: 1-2 Find the domain of the vector function. 1. r(t) = (10(1 In(t + 1), t - 2:21

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Q: 5. Use Stokes' Theorem (and only Stokes' Theorem) to evaluate ffs curlF - dS for the vector field…

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Q: 3. determine the dimension of the vector space. a. dim(R²) = b. dim(R³) = d. dim(M3x3) = e. dim(Pg)…

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Q: Find Duf at (2,−1,1) for f(x, y, z) = 3x5y²z³ in the direction given by u = (1/3, 2/3, −2/3).

A: Given function:- fx,y,z=3x5y2z3

Q: 4. Match the vector fields given below with their plots. (a) F(x,y, z) = (b) F(x, y, z) = (c) F(x,…

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Q: Consider the vector field v(z) = i+ i+4k . a) Calculate fu• dr , where l' is the straight line from…

A: We’ll answer the first part of this question since due to complexity. Please submit the question…

Q: Suppose F = (2xz+3y²) ay + (4yz²) a;. (a) Calculate Lf curl F· dS, where S is the shaded surface in…

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Q: C₁: r₁(s) C₂: r₂(t) C3 r3(u) = = = (s 2)i+ (2s² - 1)j + k 2t² i + (3t+7)j + (t+1) k (SER) (t = R)…

A: for the given vector functions for the curves C1,C2 and C3 To answer the given questions.

Q: Suppose F(x, y) = x² + y²j and C is the line segment segment from point P = (-3,-4) to Q = (-3, 1).…

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Q: Consider the function f(x, y) = 3 Find Duf(4, 3), where u = (a) v=i+j Duf(4, 3) = -1/3-1/10.3 2…

A: Since you have posted multipart question and we can answer three subparts. So, answered the first…