1. Use the parameter r =t to find a vector-valued function for the space curve represented by the intersection of the surfaces r2 + z2 = 4 and r-y = 0.

Q: Let M be the space curve given with the vector valued function R(t) = (1 + 2 sin(2πt))i + (ln t)J +…

A: The equation of tangent line of M at t=2 with  the vector valued function Rt=1+2sin2πti+ln tj+cos…

Q: 4. Find the divergence and curl of the following vector function: F(x,y, z) = ( arctan(ry),…

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Q: Let C be the curve of intersection of the paraboloid a² + y² = z and the cylinder +(y-1)² = 1 (a)…

A: Given that C is the curve of intersection of the paraboloid x2 + y2 = z and the cylinder x24 + y -…

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: Let C be the curve of intersection of the paraboloid x² + y² = z and the cylinder + (y − 1)² = 1 (a)…

A: We will use the basic knowledge of vector calculus and differential calculus to answer all three…

Q: The vector equation for the curve of intersection of the surfaces a = y? and z = y in terms of the…

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Q: Let F(x, y, z) = xye" – 2° Ja and P(1,0, –1). Let S be the surface defined by the equation xye" –…

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

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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: Show that this curve y represents the vector (1,0) as v = 1- +0- in the sense that, for an arbitrary…

A: Given that γ(t)=(t,t2)in R2, t∈[-1,1] The objective is to show that curve represents the vector…

Q: A space curve is given by the vector function r(t) = (3t, t² + 6t, t³ ). Then the unit tangent…

A: GIVEN: r(t) = {3t, t2 + 6t, t3}

Q: Let r(t) = sin t i + et j + In (t + 1) k %3D be a space curve. Which of the following vectors is…

A: See the attachment.

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: 3. Consider the curve C defined by the vector-valued function R(t) = -2):+-3)j+ cos(t – 2)k, - oo…

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Q: Surfaces z = x² + 2y², x + y = 0 X Parameter x = 5t

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Q: Find an equation of the tangent plane to the surface represented by the vector-valued function at…

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Q: Sketch the space curve. Vector-Valued Function r(t) = -ti + 6tj + 5tk 2 -2 5 0 y Z Z 2 5 Find its…

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Q: Prove that C is tangent to S at (0, -1, 1), considering that C has a vector-valued function 2 i+ 4 4…

A: Here, we are given the following equation of surface S : x3+y3+z3=xyz. Let us take f(x, y,…

Q: etch the vector field F(x, y) = xi + yj and calculate the line integral of F along the line segment…

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Q: Match the vector-valued functions (a)–(f) with the space curves (1)–(vi) in Figure 10. (a) r(t) = (t…

A: (a) Identify the graph of the vector function rt=t+15, e0.08tcos t, e0.08t sin t. From the equations…

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: 6. Given the space curve r(t) = (t, 2et, e2t), find the tangential and normal components of…

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Q: Let f(r.y) = r'y- Find a unit vector tangent to the level curve f(r, y) = 3 at its point (1,2).

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Q: 2. Consider the curve traced out by the vector valued function r(t) = (sin t cos t, cos? t, sin t)…

A: given that r(t)=(sintcost,cos2t,sint) we need to show that the curve lines on the surface of the…

Q: Sketch the space curve represented by the vector-valued function and give the orientation of the…

A: Explanation:

Q: 3. Let C be the space curve given with the vector valued function R(t) = (1+2 sin(2nt))i+ (In t)J +…

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Q: Sketch the space curve. Vector-Valued Function r(t) = −ti + 3tj + 4tk Interval [0, 1]

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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: use the parameter x=t to find a vector-valued function for the space curve represented by the…

A: Given surface x2 + z2=4 and x - y=0   and the parameter x = t

Q: Consider the vector function r : R - R° given by r(t) = (21² + t, t + 1, 1² – 2), and let C be the…

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Q: Show that if r(t) is perpendicular to the tangent vector r′(t) for all t, then the curve r(t) lies…

A: If r(t) is perpendicular to the tangent vector r′(t) for all t, then the curve r(t) lies on a sphere…

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: Choose the normal vector of the tangent plane to the surface defined by Z = e6x²+14xy-5y² at the…

A: To find: The normal vector of the tangent plane to the surface at the given point. Given…

Q: Sketch the space curve. Vector-Valued Function r(t) = -ti + 2tj + 6tk O 5 5 5 * * kk 5 Z Find its…

A: We have to sketch the vector and find the length of curve.

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

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Q: Usé the parameter x = t to find a vector-valued function for the space curve represented by the…

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Q: Let C be the space curve given with the vector valued function R(t) = (1 + 2 sin(2πt))i + (ln t)J +…

A:

Q: Let F(x, y) = (2x + y³, 2y – x4). Compute the flux oF. F.n ds of F across the circle of radius 5…

A: Given function can be written as, Region bounded by the circle of equation,

Q: ƏF dF = (dr. V)F+ -dt. at

A: Given: A vector function F depends on both space coordinates x, y, z and time t.  To Find:  To show…

Q: Suppose r(t) is a vector function in R3 Show that if r(t)is perpendicular to the tangent vector…

A: Suppose that the position vector r→t be, r→t=ft,gt,ht…

Q: 2. Compute the vector line integral: -y х — 4 Le (x – 4)² + y² dx + (x – 4)2 + y² dy where C is the…

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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& the point is (x,y)=(1,1)

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: A vector normal to the surface y-:-3 at the point (-1,-2, 2) is (a) i+2j+k (b)-i+j+2k) (e) (21 -j+k)…

A: The given surface is: xy-z=3 And, the given point is: -1, -2, 2 Since, the gradient represents 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: Match the vector-valued functions (a)-(f) with the space curves- (1)–(vi) in Figure 10. (a) r(1) =…

A: (d) Identify the graph of the vector function rt=cos3t, sin3t, sin 2t. From the function, it…

Q: Show that the vector-valued function r(t) = e−t cos ti + e−t sin tj + e−t k  lies on the cone z2 =…

A: Given vector valued function is r(t)=e-tcosti+e-tsintj+e-tk.

Q: (b) Use the corresponding formula to evaluate the line integral F. ds where F(x, y, z) = (xz, -yz,…

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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.