a. The graph y = f(x) in the xy-plane automatically has the parametrization x = x, y = f(x), and the vector formula r(x) = xi + f(x)j. Use this formula to show that if f is a twice-differentiable function of x, then |f"(x)| [1 + (f'(x))²]3/2" K(x) = b. Use the formula for k in part (a) to find the curvature of y = In (cos x), – 7/2

Q: Express the given parametrization in the form y f(x) by eliminating the parameter. - In(t), y = 5 +…

A:

Q: 1. If the parametric curve $x=f(t), y=g(t)$ satisfies $g^{\prime}(1)=0,$ then it has a horizontal…

A: TRUE Slope of the curve, when in parametric form = dy/dx = g'(t) / f'(t)

Q: find the work done by F in moving a particle once counterclockwise around the given curve. F = 2xy3i…

A: Consider F = 2xy3i + 4x2y2j We have M = 2xy3 and N = 4x2y2

Q: find the work done by F in moving a particle once counterclockwise around the given curve. F = (4x -…

A:

Q: Evaluate f(x + 2y)dx + (y – 2x)dy around the ellipse C defined by x = 4 cos 0, y = 3 sin 0,0 < 0 <…

A: Given the Integral : ∮c(x+2y)dx-(y-2x)dy ⇒Let I=∫C(x+2y,y-2x).(dx,dy) , x=4cosθ and y=3sinθ…

Q: 1. Let f(x, y) = -x2 – xy³ + 7. Find an equation for the plane tangent to f at (3, 2).

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Q: Use Green's Theorem to find the work done by F(x,y)=x(3x+y)i+xy^2j in moving a particle from the…

A:

Q: Sketch the graph of 7(t) = sin(t)i – 2cos(t)j–tk in 3-space; then find and draw the tangent vector…

A: Given, r→t=sinti^-2costj^-tk^

Q: Show that a function f(x, y, z) = (x² + y² + z²) satisfies a Laplace equation.

A:

Q: Show that the following function satisfies the two-dimensional Laplace's equation. z = In(x² + y²) +…

A: The given function is -  z = ln(x2 + y2) + 2tan-1yx We have to test whether the above function…

Q: find the decomposition of a(t) into tangential and normal components at the point indicated, as in…

A: Consider the given information rt=eti+1-tj ,  t=0   Find the first derivative of rt…

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

A:

Q: 15. What is the correct piecewise parametrization of a path C created by starting from C₁ of…

A: Introduction: The parametrization is a very significant part that is related to vector calculus.…

Q: find a parametrization c(t) of the curve satisfying the given condition. y = 3x − 4, c(0) = (2, 2)

A: It is given that,

Q: Consider the vector-field F(x, y) = y²i-xj, and let C be the arc of the parabola y = k – x² from the…

A:

Q: The circuitation of the vector field F(x, y)- (3 + xy´, 4x + xy) along the boundary of the plane…

A: The solution are next step

Q: Find a parametrization c(t) of the line y = 5 − 2x such thatc(0) = (2, 1).

A: This question comes under the category of calculus and from the parametrization of the line here we…

Q: 4. Let f(z) = 1 23 1 - and find [ f(z) dz, where C a circle of radius 2 about the origin with…

A:

Q: Find the work done by F in moving a particle once counterclockwise around the given curve. F= (4x-…

A:

Q: 1. The position of a particle in space at time t is de 3 -k: =' - 1 1+t- j+ t+1 a) lim Okelocity…

A: The position vector of the particle is r(t)=et-1ti+1+t-1tj+3t+1k

Q: { e, a| 1, The Fourier integral of f (x) = %3D A) roo w sin w-cOS w sin zwdw w2 roo sin w-CoS W sin…

A:

Q: Find the linearizations L(x, y, z) of the functions at the given points. ƒ(x, y, z) = (sin xy)/z at…

A: Linearization of a function of three variable[say f(x,y,z)] at (a,b,c) is…

Q: Suppose that (z) is entire and that the harmonic function u(x, y) = Re[fz)] has an upper bound uo;…

A: As requested following is the answer for question 12.

Q: Find a piecewise smooth parametrization of the path C. 0 st<1 (ti+tj r(t) = %3D 1<t<2 (1, 1) y=x

A: Given rt=ti + tj                             0≤t≤1                                       1≤t≤2

Q: Let f:C→C be defined by exp(-z-4) z#0 f(z) =. z = 0 Does f satisfy the Cauchy- Riemann equations? Is…

A: It is given that fz=e-1z4     z≠00           z=0 where f:ℂ→ℂ. We have to check whether the function…

Q: = Let Co be the curve formed by the intersection of the surfaces x₁ = 2x² and X3 3x1x2, and let C be…

A:

Q: Show that the function f(x, y) = e-2y cos 2.r satisfies the Laplace equation.

A: use partial derivative to check whether it  satisfy the equation or not

Q: (1) Vf (f is a scalar function) 2) ř(t) and r(s), t is time and s is arch length.

A: This is a problem of vector calculus.

Q: 15. What is the correct piecewise parametrization of a path C created by starting from C₁ of…

A: Given Path C is C1:  y=x+3,     0≤x≤4C2:  y=x2,         y=7 to y=0C3:  line connecting C1 and C2…

Q: Let ø be the solution of the Laplace equation in the second quadrant of the z-plane, satisfying the…

A: Laplace equation in two dimensions ∂2ϕ∂x2+∂2ϕ∂y2=0 Let, ϕ=Xx·Yy ∂2ϕ∂x2+∂2ϕ∂y2=X''Y…

Q: find a parametrization c(t) of the curve satisfying the given condition. y = x2, c(0) = (3, 9)

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Q: Find an equation for the tangent plane to the graph of z = xe' cos z – 2 - at the point (2,0,0). 1.…

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Q: For Problem 2, calculate the flux of the vector field F through the given surface S. 2. F = -4i+…

A: Given F→=-4i^+xj^+2zk^ and S:x+2y+z-2=0

Q: Given that the vector field F(x, y) = (2x-3y, 2y-3x) is conservative in the xy-plane. If (x, y) is a…

A: We have to find the value

Q: Show that the function f(x + iy) = Vlæ|ly| satisfies the Cauchy-Riemann equations at the origin but…

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Q: A path C is defined by the parametric equations x(t)-t, y(t) = 2t, 0≤t≤ 1, and a vector field is…

A:

Q: (10) Given that the vector field F= 2z i- 3 x j + 4 y k Evaluate f F. dr where S is the portion of…

A:

Q: Assuming that the equations define y as a differentiablefunction of x, use Theorem 8 to find the…

A: Given:           x3−2y2+xy=0,     (1,1)

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: (a) Consider the equation e'ux+2ayuy +2e*uy-sin(y) u =0, where a is a real constant. Determine the…

A: We will use basic knowledge of PDEs to answer this question.

Q: Show that the function f(x, y) = In /x² + y² satisfies the Laplace equation. %3D

A:

Q: Evaluate F.dr, where F(x,y)=[2y-3cos(x)]i + [4x +6sin(y)]j and C goes counterclockwise around the…

A: The objective is to evaluate ∫CF·dr where Fx,y=2y-3cosxi+4x+6sinyj and C goes counterclockwise…

Q: (b) Show that the function f(2) =u +iv, where S(2) = **(l+1) – y'(1-1) x² + y² = 0, z = 0 Satisfies…

A:

Q: If g is the vector-valued function defined by g (t) = (sin(2t), cos(3t)), then g' (t) = %3D

A: Given : gt = sin2t, cos3t

Q: Show that the function satisfy Laplace equation f(x,y) = In /9xy+3y² %3D

A: We need to show that the function f(x, y)=ln9xy+3y2 satisfies the Laplace equation. Now,…

Q: 1. Calculate the work done when a force F = 3xyî – y?j moves a particle in the xy-plane from (0,0)…

A: Take dot product along limit of integration and integrate

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: 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.