Consider the following functions: f1(x) = sin (x1) cos(x2) , x ∈ R 2 f2(x, y) = x ⊤ y, x, y ∈ R n f3(x) = xx⊤ , x ∈ R n (a) What are the dimensions of d fi/dx? (b) Compute the Jacobians.

Q: Let F(x, y) = x2 + x² + y2J- (a) Show that F. dr +| F. dr if C and C2 are the semicircular paths…

A: The function given to us is  F(x,y)=yx2+y2i-xx2+y2j

Q: Find the fay and fyy: f(x, y) = xe" - sin(xy) + x³y

A: fx,y=xey-sinxy+x3y

Q: Let f(x, y) be a differentiable function of 2 variables and let r(t) = 2 cos(t)i + 3 sin(t)j…

A:

Q: Find V · (V × F) if F(x, y, z) = sin xi + cos(x – 2y)j + 8zk. | V ·(V × F) :

A:

Q: 3. If G = (2x²y – x*)î + (e*y + y sin x)j+ (x² cos y)k, find: %3D дхду

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Q: Let r(t) = (sin(t), cos(t), 6 sin(t) + 9 cos(2t)). Find the projection of r(t) onto the xz-plane…

A: Given vector…

Q: 6) Consider a function f: R² → R, such that = y cos(xy). Which of the following are possible…

A: Possible candidates found using partial integration

Q: Let f(x, y) be a differentiable function of 2 variables and let r(t) = 3 sin(t)i + 3 cos(t)j. If…

A: Vector differential

Q: Find a function f such that Vf = 2e²x cos(5y)i - 5e²x sin(5y)j + z³k. f(x, y, z) =

A: Given :-  = 2e2x.cos(5y)i - 5e2xsin(5y) j + z5kNow,

Q: Q2// B :- If u+v3e"cos y and u- v=e' sin y find the Jacobain of the function alu,v at dx.yl (x.y)…

A:

Q: Img(:)=v(x.y) and g(-)=g: v(x, y) = 2x-2ry+sin xsly and g (0)=-1.

A:

Q: Given f(x, y) = sin(x2) + xy³, find Duf(x, y) where u is in the direction of (1, 2). What is Duf(-2,…

A: To find the directional derivative of given by function

Q: If v(r, y) is a harmonic conjugate of a harmonic function u(x, y) show that the function g(r, y) =…

A: Harmonic function: A two variable function ux, y is said to be harmonic, if the function is twice…

Q: |dx + N(x,y)dy = 0 x² + y,

A:

Q: What is the A B (C) D af - of f (x, y) = tan (x²y³ + 2xy)? dy [ sec 2 (x2y3 + 2xy (2 xy + 2y) 2 [sec…

A:

Q: as f(z) = e e analytic %3D

A:

Q: Find a function f such that  ∇f = 7e7x cos(3y)i − 3e7x sin(3y)j + z3k.   f(x, y, z) =

A: we need to find out the function f whose gradient is given.

Q: Given a function f(x, y, z) = ryz° +z°yln (z) – z'y? + 4, find: of (1,2, 1) = (1, 2, 1) = (1,2, 1) =…

A: Given problem based on partial derivative.

Q: sin(xz – 1) In(6+ 3xz + 4yz) = 0 determines z as function of x and y in a neighbourhood of the point…

A: Claim: at (x,y)=(1,1) for the equation  Take partial derivative with respect to x Substitute…

Q: Q: Solve on white page clean and stepwise: Let f(x, y, z) = sinxyz, g(x, y, z) = x²yz, U = x²yi –…

A:

Q: Given a function f(x, y, z) = cos (x + xy²z²), find: f#(5,1, 1) = frz (T, 1, 1) = %3D

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Q: Consider the following functions: Exercises f₁(x) = sin(x1) cos(x2), x=R² T f2(x, y) = xy, x, yern…

A: Given the functions                  f1(x)=sin(x1)cos(x2), x∈ℝ2f2(x, y)=xTy,              x, y∈ℝn…

Q: Find a harmonic conjugate v(x, y) of u(x, y) = e e sonding analytic function ƒ(z) = u(x, y)|+ iv(x,…

A:

Q: If f(x,y) = sin(2x+3y) then f yy

A: Given: A function fx,y=sin2x+3y fy is the derivative of fx,y=sin2x+3y with respect to y. And fyy is…

Q: Show that u(x, y) = e³ cos x is harmonic and find the harmonic conjugate v such that v (0,0) = 0.

A:

Q: y3 be a harm C conjugate. D[ is equal to

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Q: a(x,y) d(u,v) Evaluate the Jacobian for the change of variables X =, =u²cos(v) and y =usin (v).

A: Topic : calculus

Q: . Let X and Y be continuous random variables. The joint PDF of X and Y is 12r*y 0<y<r < 1, f(r, y) :…

A: The joint distribution of (X,Y) is given by, f(x,y) = 12x3y0&lt;y&lt;x&lt;10otherwise So,…

Q: Consider i = (1,0,0), j= (0, 1, 0), k = (0,0, 1) and, f (x, y) = (x, ²y°, sin x, cos y, xyean +3…

A: We have to make the appropriate selection  With the help of following transformation-