5) Let à = psin(4)ū, + p² Ug Evaluate f Ã. dl given that L is the contour of the following figure.

Q: Compute Div(F) and Curl(F): F--(4y-1) i+yj+(-3y) k

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Q: 1. Use an antiderivative to show that for every contour C extending from a point z₁ to a point 22, 1…

A:

Q: 1. Find a non zero unit vecor ôî that is orthogonal to u= (1,2,1) and Uz= (2,5,4) in R³.

A: Let's find non zero unit vector.

Q: B) Use Green's Theorem to evaluate F dr, where C is the triangle with vertices (0,0) (3,0) and (3,…

A:

Q: (c(2z+y), %3D ue of c in the joi F of W.

A:

Q: b) Is there a linear map T: V → W, where dim V = 3, dim W = 4, rank T = 2, and dim ker T = 1?

A:

Q: Evaluate | F· dr where F(x, y, z) (4yz cos(xyz), 4xz cos(xyz), 4xy cos(xyz)) and C is the line C 1…

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Q: Let g(r, y) = ye-, r(t) = In(t² + 1), and y(t) = vt. Find g(r(t), y(t)).

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

Q: Evaluate F · dr where C is the line segment from (2, -, 1) to (94, 1, and 2 2 F = (sin(yz) + 1)i +…

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Q: Determine la derivada direccional de f (x, y) = y In z en el punto P (1, 4) en la dirección de v = i…

A: Topic:- calculus

Q: Prove that A <m AUB where AUB = {0x | r E A}U{ly|y € B}.

A: Solution: Let us consider two sets A and B. It may be possible that A∩B≠∅. But we want to construct…

Q: Given that w = f (u, v) And u = xz. , V = dw = X az aw aw Prove that Z - ду

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Q: [2 Lo 2. and defined T: R² → R³,by T(X) = AX. Let A Find the images under T of u = |E|, and v =

A: Given a linear transformation T:R2→R3 by T(X)=AX where A= 2002

Q: Evaluate ∫ ▽f*dr Where f(x,y,z)=cos(πx)+sin(πy)-xyz and C is any path that starts at (1,1/2,2) and…

A:

Q: Find the work done by the force F = x²yi– xy²j along the paths shown from (1, 1) to (4, 2). (4, 2) a…

A:

Q: 4. Evaluate . F.ds , where C is the triangle with vertices (1,0,0), (0,2,0), and (0,0,3) oriented by…

A:

Q: 6. Let f(r, y, z) = e=² cosh y + z ar ff at the point P in the direction of :

A:

Q: Given that vec F =x^ 2 y hat i -(z^ 3 -3x) hat j +4y^ 2 hat k find the k- component of ( nabla* vec…

A:

Q: Compute (S ,g) where f(t)=1– 3r² , g(1)=t-ť'onV = C[0,1]. %3D

A:

Q: Compute F. Oszs V1-x2 –y? Let E be the hemisphere described by and F=Vz2 + x² +y² i + Vz² + x² + y²…

A: By Gauss divergence theorem:∬∂EF·nds=∭Ediv(F)dvhere givenF=x2+y2+z2( i^+j^ +k^…

Q: y Verify Euler's theorem for u = x' In

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Q: If foxJ= S** t'dt, use the fundemental theorem of culcolus, part 1, to find f'cx) fixs=

A:

Q: 5. What is UF, where F = 0? Prove the correctness of your %3D answer / computation.

A: To find- What is ∪F, where F = ∅? Prove the correctness of your answer/computation.

Q: Evaluate fs xyz dS where S is the triangle with vertices (1,0,0), (0, 2, 0), (0, 1, 1).

A:

Q: Evaluate O F.dT where +(3y %3D y + 1 and C is the triangle with vertices: (0,0), (6,0), (3, 5).

A:

Q: Prove that (aA + bB)t = aAt + bB t  for any A, B belongs to Mm× n(F) and any a, b belongs to F.

A:

Q: and EXPcess du dv as a function of u and v w = Xy + XZ + xz where x = u + v and y = u-v and then…

A: Use this formulas

Q: 1 use that transformations e Jaco Je √y=0 p√x=0 %cty octy=u dy dx = e = 1 2 and your to prove

A: An equation is given by, ∫x=01∫y=01-xeyx+ydydx = e-12. The objective is to use the transformations…

Q: tnd the kes reetor projection of anto r if

A: Given : u→=5i→+2j→ and v→=6i→-3j→

Q: Compute the discriminant D(x, y) of ƒ(x, y) = x2y2.

A: fx, y=x2y2 Partially differentiating with respect to x, we get ∂f∂x=2xy2 Partially differentiating…

Q: 1. Let TE L(V) and suppose v is in the range of T and w is in the null space of T". Show explicitly…

A: To solve we will use the definition of adjoint operator.

Q: 26. Let F(x, y) = (3x + y, a?y) and let C be the path along ry = 2 from (1, 2) to (2, 1). Calculate…

A:

Q: Compute F using, Simpson's 1/3 and Simpson's 3/8 rules and composite trapezoidal rule (for 6…

A: Simpson's 1/3: ∫abfdx=h3fa+4fx1+2fx2+4fx3+...+2fxn-2+4fxn-1+fb The value of n is 6 as 6 points are…

Q: (a) Solve the following Goursat problem for u(x,t): Utt = Uxx +4cos 2t, (t > 0, -t <x<t), u(t,t)…

A:

Q: 3. Suppose that f(1) = 2, f(4) 7, f'(1) 5, f'(4) = 3, and f" is continu- %3D | af" (x) da. ous. Find…

A: The given data is: f1=2f4=7f'1=5f'4=3  f'' is continuous  To find the value of ∫14xf''xdx.

Q: 4) Verify that Green's theorem is true for f,xy?dx – x²ydy ,where C consists of y = x² from (-1,1)…

A:

Q: dual of c, is isometric to l'.

A:

Q: Find up +Uy given that u (x,Y)s. (x,x)s

A: Given that ux, y=tanx3+mx2y2+my4m-x2-y2 To find ux+uy we follow the following steps calculate…

Q: Evaluate F. dr where C is the line segment from (6,,1) to (36, 1, and F = (sin (yz) + 1)i + (zzcos…

A: Whenever a Vector is given and we need to find line integral with two points mentioned, we first…

Q: 4. let f(x, y) = x² – xy + y² – y. find the direction u and Duf at Po(1,–1) for which Duf is the…

A: Given:- f(x, y) = x2 -xy+y2 -y To find:- Direction u and Du f at point p0 (1, -1) for which Du f is…

Q: Find the absolite marima and minima points between [o,u] gf y : * + 2dx' sunchion is porsbl.

A:

Q: The function u, takes on the following values : 1 1 1 1 1 u, = , u. + .....u, 10 1 + 3+1 + 32 +…

A: SOLUTION-

Q: Evaluate zdz from z=0 to z-4+3i along the curve C given by the line from z=0 to z=3i and then the…

A:

Q: 3.8 If H = (x – y)a, + (x² + zy)a, + 5yz a, evaluate f H dl along the contour of Figure 3.28.

A:

Q: Compute I ds over the path C : F(t) = (t, 2t,ť²) from (0,0,0) to (4, 4, 4).

A: The answer is given below:

Q: on vectors for t2 0.

A:

Q: b) Show that the following wo matricegane orthogonal tker A =-V3 -2VTO 3/T15 15 E/NTS

A:

Q: Use Green's theorem to evaluate f (x²y) dx + y²dy here C is close path formed y = x and y = x³ from…

A:

Q: Evaluate F dr where C is the line segment from (7,-,1) to (20, 1, and F (sin(yz) + 1)i + (zz cos(yz)…

A:

Fundamentals of Trigonometric Identities

Trigonometric Identities are the equalities that involve trigonometry functions and hold for all the values of variables given in the equation.