Consider the vector-valued function t cot t t +0 - Csc t Ř(t) VI+t lnt' (-2, 0, 0), t = 0. Is R(t) continuous at t = 0? note: The vector function Ris said to be continuous at t= a if it satisfies the following conditions: Ř(a) exists; • lim Ř(t) exists; • Ř(a) = lim Ř(t). t→a t→a

Q: Given f(x, y)= y sin(xy). find the gradient vector field くFx, Fy> =くy^2cos(xy), sin(xy)+xycos(xy)

A: We have to solve the gradient

Q: Let F = (2y + 2)i +4ryj + rk. (a) Show whether F is conservative. (b) Using the fundamental theorem…

A: first we show that that is conservative then which solve the given integral

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: 3. Let y be the oriented curve in R³ parametrised by t+ (t2, sin t, cos² t) for 0 <t <n. Let F be…

A:

Q: Consider the vector field F = F . dī along the curve C: F(t) = 7 cos(t)i + 7 sin(t)3, 0 <t <…

A:

Q: Consider the vector field F(x, y) = (3y² – 2xy, 6ry – x²). (a) Show that F is conservative. (b) Find…

A: According to question, Given vector field, F→x, y = 3y2 -2xy, 6xy-x2

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 Evaluate F · dr along the curve C: C 7(t) = 6 cos(t)i + 6 sin(t)j, 0…

A:

Q: Find the divergence and curl of the vector function V(x, y, z) = x cos z î + xyz j – sin z k.

A:

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

A:

Q: Consider the vector field F So Evaluate F · dī along the curve C: C F(t) = 4 cos(t)i + 4 sin(t)j, 0…

A:

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

A:

Q: Find the divergence of the vector field F. F(x, y) = 3x2i + 5y?j %3D

A:

Q: Find the work done by the vector field F = (– 4z, – 3y, – 2x) in moving an object along C in the…

A:

Q: Consider the vector field. F(x, y, z) = (7e sin(y), 2eY sin(z), 3e? sin(x)) %3D (a) Find the curl of…

A:

Q: Let C be the curve with parametrization r(t) : ,0 . Find the value of F · dr.

A:

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

A:

Q: 6. Let C be the curve in R´ given by the vector-valued function r : R → R defined as r(t) = (cos(e –…

A:

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: . Consider the vector field F = (y cos(ry), x cos(ry), – sin(z)). (a) Show that F is conservative.…

A:

Q: If a vector-valued function r(t) is continuous at t = a, then the limit r(a+ h) – r(a) lim exist.…

A:

Q: b) Given the vector field OU F(x, y, z) = 3z cos x i+ x ln y j +e° tan a k. Determine i. div F, ii.…

A:

Q: Consider the vector field F = 5 sin(x² )ï + (3x² + 2e* )j. Find the line integral F· dr, where C is…

A:

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: Let C denote the curve that starts at the point (1,0), traverses the unit circle to the point (0,…

A: Please check step 2 for the solution.!

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: Given the vector field F(x, y, z) =(z cos(xz) + ze“ )î +(3y² +1)j+(xcos(xz) + xe )k 4. (a) Find the…

A: Given F→x,y,z=z cosxz+zexzi^+3y2+1j^+x cosxz+xexzk^

Q: (4) Find a function f such that F = Vf. Consider the vector field F(x, y, z) = y cos(xy)i+(x cos(xy)…

A:

Q: Consider the line integral (Image 1) , where C is defined by the vector function: r (t) = - t2i +…

A:

Q: Consider the vector field F and the curve C below. F(x, y) = (3 + 4xy2)i + 4x²yj, C is the arc of…

A:

Q: Consider the vector field F = (r'y*, z*y") Is this vector field Conservative? Select an answer v If…

A:

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

A:

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…

A:

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(x, y, z) = In(x² + y? )i+xyj+ln(y² + 2²)k. 2x 2z +x + x2 +…

A:

Q: For the vector field G = (ye^(xy)+cos(x+y))i+(xe^(xy)+cos(x+y))j, find the line integral of G along…

A:

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

A:

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

A:

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: 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: Determine the interval(s) on which the vector-valued function is continuous r(t) = ⟨e−t, t2, tan t⟩.

A: Given: r(t) = ⟨e-t, t2, tant⟩

Q: Find the divergence of the vector field F. F(x, y, 2) = In(7x2+ 6y? )i + 42xyj + In(6y2. + 72² )«…

A:

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

A: (d) map to  (vi) (e)  maps to (iv) (f) map to (iii)

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

A:

Q: Given vector field F(x, y) = (4x³y? – 2xy³,2x*y – 3x?y² +6y³) and curve C defined by C: 7(t) = (t+…

A: Here we will find out the required value.

Q: Consider the vector field F and the curve C below. F(x, y) = (4 + 2xy²)i + 2x²yj, C is the arc of…

A: