Let C, be the straight line path z = (1 + i)t, 0 Sts 1; Cz the quarter circle z =1- cost +isin t, 0 ses }; and Ca the path ==t, (0SIS 1), 1+i(t – 1) (1

Q: Find the shortest distance, d, from the point (11, 0, –6) to the plane x + y + z = 6. = p

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Q: Evaluate I = fe(sin z + 9y) dz + (7z + y) dy for the nonclosed path ABCD in the figure. А - (0,0),…

A: Given integral is I=∫Csinx+9ydx+7x+ydy For non-closed path, integration will be…

Q: a bounded by the curve y = 3VI and the lines z = 6 and z= 8 above the z – azis is Võ+ 12v6 O 10

A: We have to choose the correct answer

Q: Let C be the closed curve indicated in the diagram. Use Green's Theorem (2,1) (o,i) (x² + y?, y³ +…

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Q: .2 Use Green's Theorem to evaluate x-ydx - xy² dy , where C is the circle x +y² = 9 with…

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Q: 5. Let C be the circle x +y} =4_which is positively oriented. Let C, be the upper half-circle C1 =…

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Q: Consider the path r(t) = (4t,2t 2 ,2lnt) defined for t > 0. Find the length of the curve between the…

A: Given the path r(t) = (4t, 2t2, 2ln t), t>0. Find the length of the curve (4, 2, 0) and (20, 50,…

Q: Evaluate I = Se(sin z + 4y) dz + (5x + y) dy for the nonclosed path ABCD in the figure. %3! A =…

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Q: Q4) Compute f z'dz, where C is the path formed by the line segment from -3 to 3 and the semicircle…

A: There is no such introduction for the solution.I have explained it within the solution.Please go…

Q: Let C be the oriented line segment starting at (0,0) and ending at (1,2). Calculate Sa F .n ds,…

A: To calculate the below line integral. ∫CF⋅nds Here Fx,y=y,−x C be the oriented line segment…

Q: Now compute [(x − 3y² + 2) ds where C' is the union of the curves C1 and C2. C1 is the line segment…

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Q: Find the length of the curve x = 5-t, for 1 < t < 2. length = y = 3-t, z = 3t 5,

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Q: 1. Determine whether the lines L₁ given by z=t, y=1 with -∞ <t<∞o) and L₂ given by z=2+s, y = 3-2…

A: Introduction: If two lines are parallel then the ratio equality is true between them. And if two…

Q: Find the shortest distance, d, from the point (4, 0, –4) to the plane x + y + z = 4.

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Q: 2人+人 2X PM the From Pi (x,,y,z,) to Pz (xz, Yz, Zz Prove that the Midpoint of the line Ssegment 12…

A: Let the 2 points in the line segment be,

Q: If C is the line segment which begins at (-2, 1) and ends at (2, 4), evaluate the line integral: e+y…

A: Given, The line integral ∫Cex+yds Where C is line segment from the point -2,1 to the point 2,4 So,…

Q: Evaluate F- dr where F(, y, z) = (1.5yz cos(ryz), 1.5xzcos(ryz), 1.5xy cos(ryz)) and C (*) is the…

A: We need to evaluate the line integral between the two points.

Q: the KX3 Determine the intersection of and plane tangem line to Cuove at t=5 . %3D

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Q: 2) Consider the oriented curve C given by C = C1uC2, where C2 is obtained by intersecting the…

A: We will first parametrize the curves C1 &amp; C2 then substitute the x,y,z values into the line…

Q: Show that the minimum distance from the origin to a point on theplane ax + by + cz = d is

A: Given: ax+by+cz=d we know that minimum distance p of a point x1,y1,z1 from given equation of plane…

Q: Find the length of the segment of the curve y = f (x) = e ^ x between the abscissa points x = 0 and…

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Q: Evaluate the line integral 2ydx + 5xdy where C is the straight line path from (1, 2) to (6, 5). Г…

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Q: Find a path c(t) that traces the line y = 2x + 1 from (1, 3) to (3, 7) for 0 ≤ t ≤ 1.

A: Given:

Q: 19. Evaluate Scx ds, where C is a. the straight-line segment x = t, y = t/2, from (0, 0) to (4, 2).…

A: Part (a)

Q: By using Green's theorem evaluate (4.ry – y') dr + (x + 3ry) dy where C is a closed curve starting…

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Q: Show that F(z, y) = (ry, z°y?) is not path independent by considering the line segment from (0,0) to…

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Q: 11. Determine whether the lines L₁ given by x=t, y = 1+ 2t, z = 2 + 3t, (with x < t < oo) and L₂…

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Q: Ue langrange multipliars to fined the Shortest distancé fam the to the pGine x+g+Z=1. point (7,0,)

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Q: Evaluate  ∫C (3y + exsinx) dx + (8x - ln(y3+2)) dy, where C is the circle x2 + y2 = 9  with positive…

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Q: Find the length of the curve given by S(x) = (a -) defined on the interval (1, 2].

A: Consider the given curve:

Q: Given that the line y =r- b is an asymptote to the ar curve y a,b €R,r = 1, then b = I-1 -1-a B 1-a…

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Q: Suppose a point X lies in the exterior of a disk, and suppose that two lines are drawn from X…

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Q: Given the line integral | (2x2° + 6y) da + (6x – 2yz) dy + (3a² z² – y²) dz. Let C be the path that…

A: Note that the following integral reduces like this :

Q: (b) Suppose C is a curve of length N, and ||F|| < M, where M is some positive number. Prove that F.…

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Q: Verify Green's Theorem for: f4y³dx-2x²dy where C, oriented anti-clockwise, is composed of the…

A: To find the integral ∮C4y3dx-2x2dy where C is the positively oriented closed curve x2+y2=9, y≥0 and…

Q: By using Green's theorem evaluate £ (Ary – y) dr + (** + 3ry) dy where C is a closed curve starting…

A: Green's theorem in a plane: If D is a region inclosed by a closed curve C then ∮C u dx+v dy=∫∫D…

Q: Let C be the line segment connecting 0 to ni. Then f. 1-e2z dz =

A: we have to find integration

Q: 2 Show that the line y = x +3x+m for all values of m. y= mx+4 will touch or intersect the curve

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Q: Let y = [3] and u = [2] to the line through u and the origin. Compute the distance from y

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Q: The area bounded by the curve y = 4r and the lines z = 2 and I=8 above the I- aris is O 240.00…

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Q: 2. Let C be the closed path constructed by joining the segment [-2, 2] on the real axis with y(t) =…

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Q: Consider the path r(t) (14t, 7t, 7 In t) defined for t > 0. Find the length of the curve between the…

A: Given - r(t) = ( 14t, 7t2, 7lnt )  defined for t&gt;0 To find - The length of the curve between the…

Q: 4. Evaluate where C is a circle of radius 2, parallel to the r - z plane and around the y-axis with…

A: Using the parametric equation of the curve C.

Q: Use Green's Theorem to evaluate -1 (y? tan a?) dx + (3.xy + 4x – In(y² + 4)) dy, where C is the…

A: The green's theorem states that "line integral over a closed curve C…

Q: Q5. Prove that the circle {(r, y), x² + y? = 1} is not homeomorphic to the interval [-1, 1].

A: Here we can prove this question using topological property connectedness.

Q: Use Green's theorem to find the k number that satisfies (k + 3) ∫C((k/2)*(x^k)*(y^2))dx + ((yx^(k +…

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Q: Consider the path r(t) = (4t, 2t², 2 In t) defined for t > 0. Find the length of the curve between…

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Q: 3. In Eercises, verify that the given point is on the curve and find the lines that are Haltangent…

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Q: Let C1 be the straight line path z = (1 + i)t, 0 < t S 1; Ca the quarter circle z = 1- cost +i sin…

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