Consider the following space curve r (t) = t²i + 2tj – v2 43/2 k %3D 3

Q: Two particles travel along the space curves r,(t) = (t, t²,t³) and r2(t) (1+ 2t, 1 + 6t, 1+ 14t),

A: Given data: The first curve is r1(t)=<t, t2, t3>. The second curve is r2(t)=<1+2t, 1+6t,…

Q: 15. If the surface S has projection R on the xy-plane, show that F.īdS = S da dy ה.F provided |ñ -…

A: Question: If the surface S has projection R on the xy-plane, show that ∬SF¯·n¯dS=∬RF¯·n¯dxdyn¯·k¯…

Q: 2 Consider the line L: y -1 +t|0, te R, and the plane T : x – 4y – z = 6. (a) Show that L is…

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Q: Find T, N, and k for the space curve r(t) = - 2e cos t)i- (2e sin t) j+ 2k.

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Q: find the work done by F in moving a particle once counterclockwise around the given curve. F = 2xy3i…

A: Consider F = 2xy3i + 4x2y2j We have M = 2xy3 and N = 4x2y2

Q: Determine whether the space curve given by r(t) = (t, t 3 , t 2 + 1) intersects the xy-plane, and if…

A: Given the space curve,  r(t)=t,t3,t2+1

Q: find the work done by F in moving a particle once counterclockwise around the given curve. F = (4x -…

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Q: If E is the solid shape bounded by x 2 0,y 20 ,z 2 0 and the plane 2x + 3y +z = 6 evaluate (x + y)…

A: The given question is solved below

Q: Use the divergence theorem to evaluate ]] F .dɔ̃ where F = xyi – ¹⁄y²j+zk and the surface consists…

A: Here, the surface consists of the three surface z=4-3x2-3y2; 1≤z≤4 on the top x2+y2=1, 0≤z≤1 and z=0…

Q: 4. Where does the curve r(t) = ti + t²j+ (3 t2)k intersect the paraboloid z = x2 +y² ?

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Q: Find a piecewise smooth parametrization of the path C. (ti + tj 0 ≤t≤ 1 r(t) = 1 y 1 ≤t≤2 [y=√x (1,…

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Q: Let C be the curve in space given by the following parameterization. r(t) = tỉ + t²j + 9k, 0 < t s…

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Q: the following velocity relations satisfy the requirements for irrotational flow. u = xt? + 2y , v =…

A: To check Irrotational Flow: For the velocity relation u and v We check equation of continuity…

Q: Sketch the space curve r(t) = i + t2j + t3k and find its length over the given interval [0, 2] .

A: If r=x(t)i+y(t)j+z(t)k then length of r over [a, b] is L=∫ab[x'(t)]2+[y'(t)]2+[z'(t)]2dt

Q: Q3. A- Use Greens theorem to evaluate the cireulation or flus for the velocity fiekd : S(x, y)=- *+y…

A: Given velocity field is  f(x,y)=1x+yi-1x+yj we have to find the flux for the boundary of square…

Q: 2. Evaluate f, z³ /x² + y² + z² dV where E is the solid hemisphere that lies above the xy-plane and…

A: Solution:- equation of sphere having centre origin (0,0) and radius=1 x2+y2+z2=1…

Q: Consider the PDE Y Urr + x Uyy + Ux – Uy x² + y? In what region of the xy- plane is this PDE…

A: compare the given equation with Auxx+Buxy+Cuyy+Dux+Euy+Fu=G A=y,B=0,C=x,D=1,E=-1 calculate B2-4AC…

Q: Find a parametrization c(t) of the line y = 5 − 2x such thatc(0) = (2, 1).

A: This question comes under the category of calculus and from the parametrization of the line here we…

Q: Use Green's theorem to evaluate (2y dx-x dy) around the closed curve C. 27 (2,8) 8. 4. C. -2 (1,-1)

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Q: Sketch the space curve r(t) = −ti + 4tj + 3tk and find its length over the given interval [0, 1] .

A: Given space curve is rt=-ti+4tj+3tk. Sketch the above space curve as shown below.

Q: Let C be the curve in space given by the following parameterization. r(t) = tỉ + t2j+ 9k, 0 <ts v7.…

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Q: Find the work done by F in moving a particle once counterclockwise around the given curve. F= (4x-…

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Q: (a) 'Find the orthogonal trajectories on the surface of the sphere x' + y? + z = 0 with the…

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Q: Find the absolute maximum and minimum values of the function f(x,y)=e** (x² +2y³) on the set D,…

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Q: Sketch the space curve r(t) = a cos ti + a sin tj + btk and find its length over the given interval…

A: Given : Space curve equation rt = acosti^ + asintj^ + btk^  ,   and  t ∈ 0,2π  ,a and b are contants…

Q: 6. (a) Show that F = (3a²y+2, x3 + 3) is conservative. (b) Evaluate Sc(3x²y +2)dx + (x³ + 3)dy where…

A: According to the problem, we have

Q: 2. Find the work done by F in moving a particle once counterclockwise around the given curve C F =…

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Q: Find T, N, and k for the space curve, where t> 0. r(t) = (2 cos t+ 2t sin t)i + (2 sint- 2t cos t)j…

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Q: Evaluate the integral [(3x – 2y + z)ds where C is the curve, C, and C, as shown. 3 y²+ z² 2 G 2* 3…

A: Given that ∫c3x-2y+zds. The objective is to find the integral value. ∫c3x-2y+zds=∫c13x-2y+z…

Q: Find T, N, and k for the space curve r(t) = - ti - (a cosh (t/a))j, a > 0.

A: Unit Tangent. Unit Normal. Curvature.

Q: Consider the one-form w = dx +g' (y) dy on R², where g : R →Ris an arbitrary smooth function. Show…

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Q: Consider the PDE y u + Uyy + u, - Uy = 2 +y? In what region of the xy-plane is this PDE elliptic? xy…

A: The objective is to find in what region of the xy-plane the pde is elliptic.

Q: Use the Green's theorem to calulate So (3x²y – xy? – 3.x²y²) dx + (x³ – 2.x³y+ x²y) dy, where C is…

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Q: 3. Sketch the space curve, r(t) = 8 cos ti +8 sin t j+tk , and find its length over the interval []…

A: Given: r(t)=8cost+8sint+tk    [0,pi/2]

Q: Consider the closed curve in the xy-plane given by x² + 2x + y* + 4y = 5. dy -(x+1) (a) Show that dx…

A: Given data:   The expression for the given function is x2+2x+y4+4y=5.   (a) Differentiate the above…

Q: Evaluate || F.ndS F = xi + yj+(z² –1)k if and o is the curved surface of the cylinder y = \4-x? from…

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Q: FIND B AND T FOR THE SPACE CURVES. r(t) = (6sin2t)i + (6cos2t)j + 5tk

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Q: Let R be the region enclosed by the loop of the t², y = t³ – 3t. parametric curve x =

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Q: Let D be the region bounded by the parabola y = x2 and the curve y = sin x, and let P represent a…

A: If the input of the function is more than one, such functions are called multivariable functions.…

Q: Sketch the space curve r(t) = ⟨4t, −cos t, sin t⟩ and find its length over the given interval [0,…

A: Consider the given space curve, rt=4t,-cost,sint use the graphing utility to graph the curve,

Q: Evaluate § F · ds, where F = 2x(x+y)i + (x² + xy + y?)j and the positively-oriented curve C encloses…

A: Here we will use greens theorem to find line integral of vector field For a vector field F→= Pi^+Qj^…

Q: Evaluate the line integral ∮C x dy−y dx where C is a circular, closed path defined by x^2+y^2=16

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Q: At what points does the curve r(t) = ti + (6t – t2)k intersect the paraboloid z = x² + y2? (If an…

A: Given, r(t)=t i +(6t - t2)k ......(1) and z=x2+y2 .......(2)

Q: Let C be the boundary curve counterclockwise of the semidisk æ? + y? 0. Then fc(-2ydx + 3xdy) 5dydx…

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Q: Evaluate the line integral y dx + x dy where C is the parameterized path x = = ² , y = t° , 3 -ydx +…

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Q: if f(x,y) =x2y÷x+y and y(t)=  (x(t),y(t)) is parametrized curve in domain of f with y(0)=(2,-1) and…

A: The first step in solving the problem is to obtain the parametric equation. The initial values can…

Q: 2. Find the parametrization of the tangent line to the space curve r(t) = (-t3, In(t), e² ) at t =…

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Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.