Let R be a vector-valued function defined by VA -1, , In(t + 1) sin 3t if t+0 (2, 3, 0) if t= 0. Give domŘ. Determine whether R is a continuous at t = 0.

Q: Determine the interval(s) on which the vector-valued function is continuous r(t) = 2e−ti + e−tj +…

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Q: Show that F is a conservative vector field and use this fact to evaluate F. dr along the given curve…

A: Consider the given function. Fx,y=x2i+y2j The compare the form of the function with the general…

Q: Find an expression for the vector field F = ⟨x - y, y - x⟩ (in terms of t) along the unit circle…

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Q: Evaluate the integral curves for the vector field F(x,y) : i – 4xy²j . О a. - 4у — х2 = C 4x2y2 Ob.…

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Q: . Consider the vector field F(r, y, 2) = (8r sin y, 4x² cos y – e"z, -e" – sec² z). %3D | F. dR…

A: For a vector field F→, the line integral about the path C is ∫CF→·dr. Let rt be the parametric…

Q: Consider the vector field F(x, y, z) = (2ry, x², 1). (c) Calculate the line integral / F - dĩ where…

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Q: Let the vector field F be given by F(x,y,z)=(y,xz,=z^2).  Evaluate its integral along the curve C…

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Q: Let F (4t +3, sin(3t), 42) Find the indefinite vector integral | F(t) dt = ( +C1, +C2, +C3)

A: Given,F=4t+3,  sin(3t),  4t2∫F(t)dt=?

Q: If a vector function v(t) = (t, Int,t-²) is given for a scalar function Ø = x In z + Jy - cos(x + 1)…

A: Given the scalar function φ=xlnz+y-cosx+1. This scalar field is to be evaluated over the path…

Q: State what it means for a scalar function y = f (x) to beintegrable on an interval [a, b].

A: Scalar integrable  function y = f (x)

Q: Define h(0,0) in such a way that h is continuous at origin       h(u,v)=ln(nu2-u2v2+nv2/u2+v2

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Q: Use the divergence of the vector field to compute F(x, y) = (tan-¹ (2x) fo tan¯¹ (2x) 4 where is the…

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Q: Consider the vector field: F(x, y, z) Show that F is not Conservative, i.e., not a gradient. sin…

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Q: Show that the vector field F = 2xyi + (x - 2 y)j is conservative and evaluate the line integral %3D…

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Q: Let r(t)= Determine the interval(s) on which this vector valued function is continuous.

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Q: Consider the vector-field F(x, y) = y²i-xj, and let C be the arc of the parabola y = k – x² from the…

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Q: Verify if the vector field F = (x° cos(4y), -x sin(4y)) is conservative and evaluate the line…

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Q: 2. Consider the vector valued funcion In(1 – t) /t+I, e-1/lt| if t +0 R(t) = t (1,0, –1) if t = 0.…

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Q: FIND THE DIVERGENCE OF THE VECTOR F GIVEN = sin (8x)i + cos (5y); + z ² k FIELD F(X, Y, 2) BY

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Q: Let r(t) be a vector-valued function such that the magnitude of r(t) does not change over time. Use…

A: if r'(t) = 0 then we can say that r'(t) is perpendicular to the function r(t) for all times t.

Q: Determine if the vector field F is conservative. F= cos x cos yi + sin x sin yj - sec |- sec zk O…

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Q: Find the divergence of the vector field F = < yx², xzª, zy³ div F =

A: f = y x2  ∂f/ ∂x = 2xy   g = x z4  ∂g/ ∂y = 0   h = z y3  ∂h/ ∂z = y3

Q: A scalar field is given by f = x2/3 + y2/3, where x and y are the Cartesian coordinates. The…

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Q: If r(t) is a vector-valued function that is continuous on the interval [a, b), then r(t) dt is a…

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Q: Show that the vector field F=2xyi +(x -2y)j is conservative and evaluate the line integral |F•ds…

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Q: Sketch the space curve. Vector-Valued Function r(t) = (5 sin t, 2t, 5 cos t) Interval [0, π]

A: Given rt=5sint,2t,5cost So, r't=5cost,2,-5sint So,…

Q: Show that F is a conservative vector field and use this fact to evaluate  F. dr along the given…

A: For a vector field of the form F=Pi^+Qj^ to be a conservative vector field, it should satisfy the…

Q: find the work done by the given vector field F along the specified curve C.   F(x, y) = yi − x j,…

A: Given, F(x,y)=yi^-xj^ where curve, x2+y2=4 W=∫F.dr W=∫ydx-xdy from curve, dx=-dy W=∫xdx+ydy limit of…

Q: Is the following vector-valued function continuous at t=0

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Q: What is the divergence of the vector field F of F(x,y) = x² i + 2y²j?2 div F = 2x + 4y div F = 4x +…

A: Given: F→=x2i+2y2j

Q: Find the net outward flux of F =axr across any smooth closed surface in R, where a is a constant…

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Q: Let F be a vector field and Ca curve with parametrization 7 (t) where a <t<b. F. dr = | (cos(t³),…

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Q: Test the vector field F to determine if it is conservative. F= -cOS X cos y i+ sin x sin y j - sec 2…

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Q: 3. Let f : [a, b) -R be integrable. Show that f(t)dt = lim,b Sa f(t)dt. %3!

A: Recall the results mentioned below

Q: Find the line integral of the vector field F= over the curve C which is the quarter of the unit…

A: We have to find line integral from point (0,1) to (1,0)

Q: If F = Pi+ Qj+Rk is a vector field on R and P, Q. and R have continuous second-order partial…

A: Given:F=Pi+Qj+Rk is a vector field  in ℝ3To show that:∇.curl F=0

Q: Find g: R R, continuous in [0. ) and position in (0, ) satisfying g1) %3D )dt

A: Here we have,  g:R→R, continuous in [0,∞) and position in (0,∞) Also given g(1)=1 and…

Q: Use the divergence of the vector field to compute F(x, y) = (tan-1 (2²)) i + (xy-2 +82²) j 4 fo…

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Q: Let be the vector field defined on the plane by F(x, y) = (x¹+3y, 6x - siny).

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Q: Show that F is a conservative vector field and use this fact to evaluate F. dr along the given curve…

A: A vector field F is conservative if Curl F =0 that is  ∇×F=0∇=∂∂xi+∂∂yj+∂∂zk where ∇→the operator is…

Q: Find the divergence of the vector field F at the given point. F(x, y, z) = xyzi + xz²j + 4yz²k; (3,…

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Q: Consider the vector field F and the curve C below. F(x, y) = x2y³i + x³y?j, C: r(t) = (t2 – 2t, t2 +…

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Q: the vector field F is shown in the xy plane. y 21 2 Is div F positive, negative, or zero? Positive…

A: Given : The vector field F is shown in the xy plane :   We have to determine that : Is div F…

Q: Let r be a di¡erentiable vector function of t. Show that if r # (dr/dt) = 0 for all t, then  r  is…

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Q: The vector field F(r, y) = (2xy +y) dx + (æ + 2xy) dy is conservative. %3D The function f(x, y) = x²…

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Q: Use the divergence of the vector field to compute F(x, y) У · (tan-¹ (2x))i + (xy − 2 + 8x²) j - = 4…

A: we are given the vector field as  F(x,y)=tan-1(2x)4i+(xy-y2+8x2)j to compute…

Q: Consider the vector field F (x, y) = i-zj and the real-valued function = tan-(#). One has x²+y² f(x,…

A: There is no error(s). All calculations are right. Since a vector field  F¯ is conservative if and…

Q: Let F = (t+ 4, sin(4t) , t°) Find the indefinite vector integral F(t) dt = ( +C1, +C2, +C3)

A: Given, F = (t+4, sin (4t), t³) ∫F(t) dt = ∫(t+4, sin (4t), t³) dt ∫F(t) dt =( ∫(t+4)dt , ∫sin(4t)dt…

Q: Q3 Show that the vector field F= 2xyi +(x -2y)j is conservative and evaluate the line integral |F•ds…

A: Solution

Q: Evaluate F Î ds for the vector field F(x, y) = yì – xj counterclockwise along the circle x² + y² = 1…

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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.