Consider the vector function $(t) = te“ i+ cos(t)j + k.Deter mine : a) ds/dt b)

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

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Q: 2. Let f(x,y) = t Find the direction of a unit vector so that Daf is minimal at point (0, 1). %3D 3…

A: To find the direction of a unit vector u so that the directional derivative of f at point (0,1) in…

Q: If the electric potential at a point (x, y) in the xy-plane is V(x, y), then the electric intensity…

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Q: Sketch the space curve represented by the vectorvalued function                  r(t) = ti + (2t −…

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Q: Use Green's theorem to calculate the work done by force F on a particle that is moving…

A: This is a greens theorem problem

Q: . Sketch the vector field F(r, y, z) = ri+ yj+ zk.

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Q: Find the flux of the vector field F(x, y, z) = ri across the outward oriented sphere %3D

A: In this question, we find the flux of the vector field f(x,y,z)=xi The surface is a sphere of radius…

Q: given the vector function f(t) = (t, e', –2cos 2t) determine a)df/dt f(t)dt

A: The vector function is given by ft = t2, et, -2cos2πt To evaluate: a dfdtb ∫01ftdt

Q: For y'= Aj where A is given below %3D A= find the right hand side vector F(t, ) needed to apply…

A: Given y'→=Ay→ Here A=1245 We have to find right hand side vector F→(t,y)→.

Q: Find the flux out of the unit shpere for the vector field F(2,4, ;) = 1ā+3+K). %3D r answer =

A: Flux of F through the closed surface S is ∫∫sF→·n→dsS is the unit sphere x2+y2+z2=1Letϕ:…

Q: b) Given the vector field SUTM F(r, y, 2) = 3z cos ri+rlnyj+e* tan rk. Determine UTM i. div F, ii.…

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Q: 3. fimdia a.A. vector.artheogenal.te. xv.and. Txw. b. A vectore.or.thogenal.b. und. SA…

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Q: Find a vector function that represents the cur ve of intersection of the paraboloid z a + mnd the…

A: Given that, the paraboloid is z=x2+y2 and the cylinder is x2+y2=22. The projection of the of the…

Q: Find the work done by the force F = 7i - 3j in moving an object from the point (2, 2) to the point…

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Q: Find the flux of the vector field V(x, y, z) = 6xy²i + 7x<yj + z°k out of the unit sphere.

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Q: A vector field v is expressed in spherical coordinates as v(r, 0, 0) = 2r sin cos o e, +r cos o cos…

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Q: - Given the vector fields Å = yz az – 3x a, +x²y a, B = 3 a, – 4xy a, +y² az Petermine a unit vector…

A: Given that: A→=yzax-3xay+x2yazB→=3ax-4xyay+y2az

Q: Find the flux of the vector field V(x, y, z) = 9xy²i + 9x²yj + z³k out of the unit sphere.

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Q: Find the work done by the force F = 3i − 9j in moving an object from the point (3, 3) to the point…

A: To find work done.

Q: if A=i+j+X, B=zi-j+3ksfind vector normal to B and in plane of A and B- ans. $2(3i+9j+x)

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Q: Show that ¥(x,t)= Asin(wt+kx) describes a wave motion

A: Given: ψx,t=Asinwt+kx

Q: 问题的回答系 极性问题是预期答案等 于 是和否的问题(因此, Given the vector field F(r. y, z) = 3: cos.ri+a lny j + e° tanr k.…

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Q: Find the work done by the force F = 5i + 25j in moving an object from (0, 0) to (1, 6).

A: Using formula of work done,

Q: A vector field v is expressed in spherical coordinates as v(r, 0, 6) = 2r cos 0 e, – r sin 0 eg.…

A: For the solution follow the next step.

Q: Calculate the flux of the vector field (x°, y), out of the annular region between the a + y? = 16…

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Q: Q1:- Find the domain of the following vector fumctions:- (a) f(t) = (cos nt)i - Ln(t)j + Vt-2k (b)…

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Q: Determine a direction derived from the point P and in the direction of the versor vector → informed…

A: Given:  Let f be a differentiable function at the point (a, b). Then f has a directional derivative…

Q: Given a differentiable vector function r(t) defined on[a, b], explain why the integral ∫ba||r'(t)||…

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Q: Sketch the vector field: F(x, y) = (1, 2y) Carefully draw at least four vectors in each of the four…

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Q: Let &lt;p = In r, where r = J x 2 + y2. Express V &lt;p in terms of the unit radial vector er in R 2…

A: To Determine: express ∇p in terms of the unit radial vector er in R2. Given: let p=ln r  , where…

Q: Show that the work done by a constant force field F = a i + b j + c k in moving a particle along any…

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Q: Find the work done by the force F= 6i – 2j in moving an object from the point (1, 7) to the point…

A: Work done by force F→ in moving object from point P to point Q is given by F→.PQ→ , where PQ→ = OQ→…

Q: A vector field representing wind speeds is given by v = æy³ i + 3x³y³j. %3D A walker can take one of…

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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: ) Show that the vector field, F3 yz £ + zx ŷ + xy 2 c

A: Given: The vector field is, F3 = yz x^ + zx y^ + xy z^

Q: Find the work done by the force field F in moving an object from P(-9, 7) to Q(3, 3). 2.x. F (r, y)…

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Q: Consider the curve r(t) = (t, 3 cos t, 3 sin t) for t 2 0. Is the integral o r(t)dt a number or a…

A: Given :   r(t) = ( t , 3cost , 3 sint )

Q: The spherical equivalent of the vector B = yi + (x + z)j located at (-2,6,3) is given by

A: The variables of the spherical coordinate system r,θ,ϕ are given as.…

Q: Let W be the union of the second and fourth quadrants in the xy-plane. That is, let W= Complete…

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Q: Find the Duf of f(x, y,z) = z In(x+ y) + e3y on the vector u = 3i -j-4k at Po(1,0,1)

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Q: (b) Solve the electric flux through a paraboloid z = x² + y² – 16 that lies below plane z = 0. It is…

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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: Find the work done by the force field F in moving an object on the line segment from A to B, where…

A: Given, Fx,y=2y32 i +3xy j To find the work done by the force field F in moving an object on the line…

Q: Obtain the work done by the force field F(x, y, z)=xi - z³j+zek in moving a particle along a curve…

A: Given  Fx,y,z=xi-z3j+zeyk               =x,-z,zeyr(t)=sin ti+2etk                  0≤t≤π

Q: For scalar functions u and v, show that B = (Vu) × (Vv) is solenoidal and that 1 A = (uVv – vVu) is…

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Q: Find the constants a, b, c so that the vector à = (-5x – 6y + az)i + (bx + y – 2z)j + (-3x + cy –…

A: Given the vector  A→=-5x-6y+azi→+bx+y-2zj→+-3x+cy-4zk→ Since the vector is irrotational the curl…

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.