1. Determine whether the vector-valued function 't² – 9 v - F(t) = t - 3 V3t – 1, t - 3 if t < 4, t # 3 In(4 – t), (6, 2, 1) if t = 3 is continuous at t = 3.

Q: Q2. If F = (x² - y² + 2xz)ī− (2xz − xy + yz)j + (z² + x²), find the divergence of curl F and curl of…

A:

Q: Let r be a differentiable vector function of t. Show that if r . (dr/dt) = 0 for all t, then | r |…

A:

Q: Q7:- Find the limit of the following vector functions:- (1) Lim (t²i+ 2tj) (2) Lim(+1i, (3) Lim (ti…

A: We have to find the limit of the vector function.

Q: Show that the vector function r defined by r(t) = ƒ(t)i + g(t)j + h(t)k is continuous at t = t0 if…

A:

Q: Consider the vector field F = (5x + 6y, 6x + 4y). F - dr along the curve C given by r (t) = (t², t*)…

A: ∫CF→·dr→=∫abF(r(t)) r'(t) dr

Q: 10. For a vector field F = <2xy + y³, x² + 3xy²+ 2y7 find a function f such that F = Vr.

A:

Q: Find the gradient vector field of f(x, y, z) = 9x² + 8y* + 6z3

A:

Q: 1) Given the vector-valued function r(t]=e",-te"",tlnt) Evaluate lim r(t) and Evaluate lim r(t) b-…

A: Consider the given vector-valued function

Q: 5. Let F be the gradient vector field of the function f (x, y) = 4xy + 3y". Evaluate JaF. dr, where…

A:

Q: 2. Consider the vector-valued function R(t) = { Show that is continuous at t = 1. (el-¹, 1-², In (2-…

A:

Q: Express the following as a vector valued function 7 (t). 1. The line L defined by y+3=3x and z+1 =0…

A: We will express the line L defined by y+3=3x and z+1=0 hyperbola -x2-2x+2y2+4y=-2 as a vector valued…

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: 41. Let r be a differentiable vector function of t. Show that if r•(dr/dt) = 0 for all t, then |r|…

A:

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: Identify the vector-valued functions from the list below. O 6 (y) = y6 – 13e k (q, r, s) = In (13qr)…

A: A vector valued function is a function whose domain is a subset of real numbers and the range is a…

Q: 1. Let F(x, y) = 1 0 if x + 2y ≥ 1 if x + 2y < 1 Does F defines the DF in the plane?

A:

Q: Sketch the space curve. Vector-Valued Function r(t) = (2 sint, 7t, 2 cos t) 1.0 0.5 0.0 0.6 -0.5]…

A:

Q: 2. Use the limit definition to find the directional derivative of the given function f at the given…

A:

Q: Consider the vector-valued function [f(x, y) xt – xy² [g(x, y) 2 Vx + y³x² [f(x, y) Which of the…

A: Find the Jacobian of the given matrix

Q: Consider the vector-valued function (t - 2)² In(t – 2), sin(t – 2) 12 - 4 12-3t+2 12 +t-6 Find dom R…

A: Here, the domain of R(t) is nothing but an intersection of the domains of each term. Domain of…

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

A:

Q: 2. Consider the vector-valued function 12t3 – 72t2 + 96t 15t – 60 6t – 24, if t > 0, t + 4, Ř(t = 4…

A:

Q: Find the divergence of the field F. F = 8x 5 i + 3y 5 j + 2z 5 k 8x4 + 3y4 + 2z4 O 40x4 + 15y4 +…

A:

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

A:

Q: Assume that z = ƒ(x, y), x = g(t), y = h(t), ƒx(2, -1) = 3, and ƒy(2, -1) = -2. If g(0) = 2, h(0) =…

A: We use the partial derivatives to find dz/dt  Then plug x=g(t) and y=h(t)

Q: 2x Find the gradient vector field (F(, y, z)) of f(x, y, z) = y cos F(2,y, 2) =

A:

Q: Find the gradient vector field (F(r, Y, 2)) of f(x, y, z) = In(4x + 6y + 22) . F(7, y, 2) = (

A:

Q: 4. Determine whether or not the vector field F(x, Y, z) = (-xye®, -xe + e° + 2y, 2z) is…

A:

Q: 1 2. Determine the numbers at which the vector-valued function R(t) = t²i+ ln (t-1)j + 1 ) ³ + + = =…

A:

Q: Consider the vector-valued function t² – 3t + 2 Ř(t) = ( (t – 2)² In(t – 2), sin(t – 2) t2 – 4 | t2…

A: According to question, Vector valued function, R→t = t2-3t+2t2+t-6, t-22 lnt-2, sint-2t2-4.

Q: Is the vector valued function continuous at t=3? Defend your answer. I² -9 t- 3 V3t – 1, In(4 – t),…

A: Topic = Continuity

Q: Find the divergence of the vector field F. F(x, y, z) = In( 7x²+ 4y² )i + 28xyj + In( 4y2 + 5z2 )k

A:

Q: I. Consider the vector-valued function In(2t + 1) Ŕ(t) = ²₁+ î+ t 1. Determine the domain of R. 2.…

A: This is a problem of Vector Calculus.

Q: 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.…

A:

Q: 19. (a) Evaluate the line integral F. dr, for F(x, y) = ex−¹ i + xy j and C is given by r(t) = t² i…

A: (a)F(x, y) = ex-1 i + xy j   And C: r(t) = t2i +t3j , 0≤t≤1.⇒drdt = 2t i + 3t2j and F(r(t)) = et2-1…

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: Suppose that z = f(x, y) is differentiable at the point (1024, 256) with f-(1024, 256) = 4 an dz…

A:

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

A:

Q: The divergence of a vector field F = 3x² yz? î + 2yz ĵ – xz² k at he point (1,0, 1) is -2 O 2 O 1

A:

Q: Find that F = 12x²yi + (4x + 8yz²)j + 8y² zk is conservative, and find a function f such that F =…

A:

Q: Suppose that z = f(x, y) is differentiable at the point (1024, 1024) withf,(1024, 1024) = 5 and…

A:

Q: Suppose directional derivatives of the scalar function f(P), at a point P = (x, y), in the…

A:

Q: 5. Consider the vector valued function: 7(t) = (1,t² – t, sin 2t) (a) * Compute 7'(t) (b) , Compute…

A: Solution

Q: 3. Evaluate |F. dr where C is the curve F(t) = (t, t°, t“), 0<t< 1 and the vector field F(r, y, z) =…

A:

Q: Sketch the space curve. Vector-Valued Function r(t) = (4 sin t, 7t, 4 cos t) __4_2 1.0 0.5 0.0 0.0…

A: Given data÷   rt=&lt;4sint,7t,4cost&gt;   For curve option A is correct one.

Q: 1. Find R(t) which is supposed to be a differentiable vector fund t > −1 with R′(t) = (1+' 1+²,1 +…

A: Given- R'→(t)=11+t,11+t2,1+t2 and R→(0)=1,1,-1 Here we will use the following integration formulas-…