Consider the one-form w = dx +g' (y) dy on R², where g : R →Ris an arbitrary smooth function. Show that for any curve C C R² whose endpoints Pand Q lie on the x-axis, | Scw is equal to the distance between P and Q.

Q: Let y be the curve with vector form r(t) = (1/(t+ 2), sin(sin(sin t)), e-"). a) Determine whether…

A: The parametric curve is r(t)=⟨1t+2, sin( sin( sin t)),e-t⟩. x(t)=1t+2, y(t)=sin( sin( sin t)),…

Q: Evaluate the line integral y dx +xdy; C is the curve x = 4t, y = t³ – 1, 0sts 1. Jy dx + x*dy (Туре…

A: Consider the provided question,

Q: Suppose at t minutes, r(t) meters is the radius of a circular oil spill from a ruptured tanker and…

A: The given data is: rt=4t2+20,    if 0≤t≤216t+4,     if t>2 To prove that r is continuous at 2.

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Q: Find the function f if. Vf-(3a²y + y², a² + 2xy), Vf dr with and without the Fundamental Theorem of…

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Q: Show that at any point (x,y) on the curve y=bex/a ,the length of the subtangent is constant and the…

A: The given curve is y=bexa.

Q: Evaluate the line integral along the curve C. (xz+2) ds, C is the curve r(t) = (-6-2t) i + tj- 2tk,…

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Q: Determine the exact area between the x-axis and the curve defined parametrically by ¤(t) = ;t2 1…

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Q: Suppose F(x, y) = (6x + 5y)i + (5x + 4y)j. Evaluate the line integral for each of the given paths,…

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Q: Evaluate fpxV47x² + 47y² dA, where D is the shaded region enclosed by the lemniscate curve r? =…

A: To find the value of the double integral         ∫∫Dx47x2+47y2dA where D is the region enclosed by…

Q: Evaluate the line integral for the given curve C. S-ds, C: r(t)= (t',t*),1<t<2 cy ||

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Q: Evaluate the line integral F(x, y) = xyi – y²j where the curve C is repre sented by R(t) =t²i+t³j…

A: Function : Fx, y = xy i - y2 jRt = t2i + t3j from (1,1) to (4, -8)

Q: b. Evaluate the line integral ycos(xy)dx +(xcos(xy)+x)dy where R is the circle x'+y =9 traversed…

A: Solution Given b). ∫y cos(xy)dx+(xcos(xy)+x3) dyevaluate the line integral

Q: Let - (y dx – x dy) W = (x² + y²)ª be a one-form on U =R² \ {(0,0)}, and o : V → U be the smooth…

A: Given: Let us consider the given, ω=2x2+y24ydx-xdy be a one-form on U=ℝ2\0,0 and ϕ:V→U and the…

Q: Find the absolute extrema for f(x,y)=4-x^2-y^4+1/2y^2 over the closed disk D:x^2+y^2 is less than or…

A: If f(x,y) is function of two variables then the points which satisfyfx=0fy=0 are known as a…

Q: Let F = Vf, where f(x, y) = sin(x - 3y). Find curves C, and C, that are not closed and satisfy the…

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Q: 2. Let L be the line tangent to the space curve represented by r(t) = (t² – In(t))i+(e²-t – t)j+…

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Q: Find an equation for the level surface of the function f(xy.z) = x+ ey+z that passes through the…

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Q: Let C be the curve in space given by the following parameterization. r(t) = tỉ + t2j+ 9k, 0 <ts v7.…

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Q: B) Evaluate the line integral F-dr. where F (x, y) = x*y²i+x²y°j and C is the curve r (t) = (t³ +…

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Q: Suppose that f(x, y) = xy and that C is the curve 7 (t) = (t², t³), where 0 <t <1. Fill in the…

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Q: Show that (d/dz)ž does not exist anywhere, i.e., f (z) =z is non-analytic anywhere.

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Q: Suppose C is the curve r(t) = (6t,t°), for 0sts2, and F = (3x,3y). Evaluate F•Tds using the…

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Q: Let F = e'i+(xe' -e')j- ye'k and cbe a smooth curve defined by Then the value of the line integral…

A: The formula is  ∫F.dr=∫F(r(t)).r'(t)dt Here…

Q: Consider the following line integral | (ædx + ydy + (xz – y)dz) along a curve C given by r = t2²i+…

A: We will find out the required value.

Q: Find T, N, and k for the space curve r(t) = - ti - (a cosh (t/a))j, a > 0.

A: Unit Tangent. Unit Normal. Curvature.

Q: Evaluate the line integral ∫Cx2ds, where C is a the curve given by the equation y=f(x)=lnx, 1≤x≤e.

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Q: Let G(x. y)=x*yi+(y+xy*)j. Evaluate the line integral [G-dr, where Cis the curve x= ť and y=t* for…

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Q: Consider the one-form w = x dx +y dy + z dz on IR3, and the smooth function o : R → R³ given by $(t)…

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Q: Show F(x, y) = (4xy³ + 6)i + (6x²y² + 2e²¹)j LF is conservative by finding a potential function f…

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Q: Q7. Use Green's theorem to evaluate fy² dx + 3xy dy, where c is the boundary of the semi annular…

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Q: Find the absolute extrema for f(x, y) = (x² + 2y²)e-x²-y' over the closed disk D: x° + y° < 4. 2.5 2…

A: Given a function of x and y. We need to find the absolute extrema ( maxima and minima) over a given…

Q: Evaluate the line integral F. dr, where F (x, y) = (3+2xy) i+ (x² – 3y²) j and C is the curve given…

A: Given problem:-

Q: Evaluate f, x*dz + z*dy + y*dx where C is the curve x = t,y = t, z = t? from t= 0 to t= 2.

A: x = t2 ,  y=t2, z=t2 dx = 2t dt,  dy= 2t dt, dz = 2t dt  Putting values in the integral: ∫02t4(2t…

Q: Find the tangent plane for the level surface f(x,y,z) = 7x³ + cos (4y) + In (4z) = c at the point…

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Q: Evaluate the line integral T, (“+y) dx + 4 ry dy where L is the path, -> r) = (3 t, 2) with 0 <t <1.…

A: We evaluate given integral

Q: For F(x, y,z)=(6xsin y+10xz' )i+(3x cos y-4ye"-2y ze" ) j+(-2y'e* +15x²z² )k , find | a function f…

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Q: Let F = Vf, where f(x, y) = sin(x - 3y). Find curves C, and C, that are not closed and satisfy the…

A: There are so many curves along which the integral values are specified value.

Q: (a) If A = (4xy – 3x²2²)i + 2x?j – 2x°z k, prove that A dr is independent of the curve C joining two…

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Q: Let F = Vf, where f(x, y) = sin(x – 9y). Find curves C, and C, that are not closed and satisfy the…

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Q: Use Green's theorem to evaluate the line Integral for the curve C given in the figure. [2y dx + x…

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Q: Evaluate the line integral ∮C x dy−y dx where C is a circular, closed path defined by x^2+y^2=16

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Q: F(x, y) = xi + yj C: r(t) = (9t + 2)i + tj,      0 ≤ t ≤ 1

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Q: Use Green's theorem to evaluate the line integral for the curve C given in the figure. [8y dx + 7x…

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Q: Suppose F(x,y)=(4x+6y)i +(6x+4y)j. Evaluate the line integral for each of the given paths, which are…

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Q: Show that the curve r(t) = (ln t)i + (t ln t)j + t k is tangent to the surface xz2 - yz + cos xy = 1…

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Q: Suppose C is the curve r(t)= (3t,3t), for 0 st s2, and F= (8x,5y). Evaluate F T ds using the…

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Q: Evaluate the line integral y dx + x dy where C is the parameterized path x = = ² , y = t° , 3 -ydx +…

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Q: Evaluate the line integral. e ds where C is the curve given by x=2t, y=3t, 0<t<2

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