5. If a vector function F depends on both space coordinates (x, y, z) and time t, show that dF = (dr · V)F + .

Q: Prove that if u is the vector function with the constant value C, then du/dt = 0.

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Q: 1. Use the parameter r =t to find a vector-valued function for the space curve represented by the…

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Q: If a particle moves in the xy-plane so that at time t > 0 its position vector is (In (t2 + 2t) ,…

A: As we know that the expression for the velocity is; v=dstdt Given position vector is:…

Q: 1) If W = f(x, y,z) is a differentiable and x,y and z are differentiable functions of (t). By using…

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Q: Find llFll and sketch several representative vectors in the vector field.

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Q: 4. Is the vector field F(r, y, :) = (yz,rz, ry) conservative? Justify your answer.

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Q: H.W: Let V = {(x, y): x 20} with standard operation on R2. Is Va vector space?

A: A vector space is a set V that satisfies the following 8 conditions. Here two operations "+" and "."…

Q: Consider the vector function r(t)=⟨t,t^2,t^4⟩ Compute r′(t)=⟨  ,  ,  ⟩ T(1)=⟨  ,  ,  ⟩ r′′(t)=⟨  ,…

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Q: Q3) Compute the work done by the force field F(x,y,z) = (3x, xz, y) on the curve C, where C is the…

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Q: If an object follows the path of the two-dimensional vector-valued function p(t)=(16−cos(t),…

A: Given p(t)=(16−cos(t), 0.6πt−0.15t2−3)

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

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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: Compute the line integral of the vector field F=⟨3zy-1 ,4x,−y⟩ over the path c(t)=(et,et,t)for…

A: To Compute the line integral of the vector field F=⟨3zy-1 ,4x,−y⟩ over the path c(t)=(et,et,t)for…

Q: 3. Find a vector function that represents the curve of intersection of the paraboloid z =. and the…

A: To find the vector function that represents the curve of intersection of the paraboloid z=x2+y2, and…

Q: 9. Sketch the vector field F(x, y) : :(yi — х). Vx? + y?

A: Consider the given filed, Fx,y=1x2+y2yi−xj Further simplify. Fx,y=yx2+y2i-xx2+y2i=〈yx2+y2,-xx2+y2〉

Q: Theorem 7.5.4. The curvature of a smooth space curve defined by the vector valued function R(t) is…

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Q: 17. If a vector function v(t) = (t, ln t, t-²) is given for a scalar function Ø = x Inz + √y-cos(x +…

A: We have to determine the line integral.

Q: If Ū is a unit vector and Di f(x, y) = 0 for all points (x, y) then f is a constant function..

A: We have to determine whether the following statement is true or false : If U→ is a unit vector  and…

Q: How much work is required to move an object in vector force field F = (yz, xy, xz) along path r (t)…

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Q: find laplace of f(x,y,z)=2x3y2z2

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Q: Show that the vector field F (x, y, z) = xz i + xyz j y 2 k is not conservative.

A: Consider the following vector: Fx,y,z=xzi^+xyzj^+y2k^ Find the Curl of F→:…

Q: f f(x, y, z) = xy: on of vectorî - %3D

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Q: 4. Is the vector field F(x, y, z) = (yz, xz, xy) conservative? Justify your answer.

A: Calculate divergence of given vector field. If divergence is zero then field is conservative.

Q: Compute the vector assigned to the pointP = (−3, 5) by the vectorfield:(a) F(x, y) = (xy,y − x)(b)…

A: To Determine : (a) F(x, y) = (xy,y − x)(b) F(x, y) = (4, 8)(c) F(x, y) = 3x+y , log2(x + y) Given:…

Q: Find the gradient vector field (F(x, y, z)) of f(x, y, z) = In(4x + y + 62) . F(1, y, 2) =

A: we have to find the gradient vector field of f

Q: Compute the line integral of the vector field F=⟨3zy−1,4x,−y⟩ over the path c(t)=(et,et,t)for −4≤t≤4…

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Q: 6) (From Hardcover Book, Marsden/Tromba, Vector Calculus, 6th ed., Section 4.1., # 20) Show that, at…

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Q: 8) Given z =f (x.y) =1- (x² +4y2) ,Find : a)The vector function, 10 normal to the surface of f at…

A: This is problem of vector calculus.

Q: Which of the following vector functions represents the curve of intersection of the cylinder x2 +y =…

A: Given Data: The equation of the curve is: x2+y2=4 ......1 The equation of the plane is: x+z=3…

Q: Consider the curve C in R3 whose parameterization is given by: eq. in image If is there a point on…

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Q: If vector T = (x+ y+1)i+ j– (x +y)k then T.curl(T )is

A: We are given vector T.

Q: For the vector field in three-dimensional space F = (4x°y – sin(x – 2))i+ e² v#j– yzk Evaluate…

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Q: Show that the following vector field is conservative and find the potential F(r, y, z) = (2ry, x² +…

A: The answer is given as follows.

Q: -dt. dF = (dr · V)F+ at

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

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Q: Show that whether if the co-vector w can be written as dø differential? w = y. (x +). dx + x. (+).dy…

A: Given:  w=y·x+y2· dx+x·x2+y·dy....(1)

Q: Suppose r(t) is a vector function in R3 Show that if r(t) has a constant speed, then r′(t) is…

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Q: Find the point on the graph of z = 3x2 + u? at which vector n (24, 2, –1) is normal to the tangent…

A: Gradient of the surface denote the normal direction.

Q: 2. Sketch the vector field F(x, y, z) = xi + yj + zk.

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Q: Suppose that C is the graph of a smooth vector-valued function r(s) parametrized by arc length with…

A: Introduction: The tangent vector to a vector valued function rt is formulated by the equation r't…

Q: o) Find the flux of the vector field F (x,y,z) = 3xi – yj + 2zk; across o where o is

A: The given question belonging to the vector calculus  in which we have to find the value flux of the…

Q: Find the work done by the force F (x; y) = xyi-(y-x)j over the straight line from (1,1) to (2,3)

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Q: Let ƒ(x,y) = x²e"9. Give the unit vector that points in the direction of maximum increase along the…

A: Given that: f(x,y)=x2exy&amp; the point is (x,y)=(1,1)

Q: Let, T:R' → R';T(x, y,z) = (x – y,2x + 3y + 2z,x+ y+2z) . verify that [T][v], = [T (v)]; for any…

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Q: B/ show that if u(t) = d cos t + b sin t, where a and b are constant vectors |du d'u dt dt then…

A: B. Given: u(t)→ =a→ cos t+b→ sin t  where a→  and b→du⇀dt=-sin t a→ + cos t  b→d2u⇀dt2=-cos t a→ -…

Q: Show that on the saddle surface z= xy the two vector fields (Vī+x² ± i+ y°, yvī+x² ±xvI+y°) are…

A: Let f(x,y,z)=xy−zf(x,y,z)=xy−z , (a,b,c)∈the saddle surface, and calculate the total derivative…

Q: 9. Evaluate the line integral ſo F · dr, where F(x, y) = (ry, 3y?), and C is is given by the vector…

A: We have to integrate F • dr over the curve C given by the vector function r(t) = ( t4, t3 ) for t in…