Let W be the work (against the Sun’s gravitational force) required to transport an 80-kg person from Earth to Mars when the two planets are aligned with the Sun at their minimal distance of 55.7 × 106 km. Use Newton’s Universal Law of Gravity (see Exercises 35–37 in Section 6.5) to expressW as an integral and evaluate it. The Sun has massMs = 1.99×1030 kg, and the distance from the Sun to Earth is 149.6×106 km.

Q: Find the work done by a force F = 6ỉ + 5j– 3k when a particle moves from point P(-3, 4, 5) to point…

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Q: Show that for any scalar, vector or tensor field y and any material volume V: d. prpdV = dt v Dt AP-

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Q: Compute the work done by the force field F = in moving a particle along the line segment from (1, 1)…

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Q: Match the vector field with its graph. F(x, y) = yi 址 四 場 x Q 6 電

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Q: In Exercises 5-40, evaluate the integral using FTC I.

A: Given: ∫0π3cost dt

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Q: In Exercises 85-98, use the Change of Variables Formula to evaluate thedefinite integral.

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A: Given a vector field as F=2xyx2+1-2x,lnx2+1-2 We need to find the scalar potential from which this…

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Q: Find the conservative vector field for the potential function by finding its gradient. (x, у, 2) %3…

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Q: Show that the field F = (cos x)i + (- y sin y)j + (z3)k is conservative and Önd a potential function…

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Q: Evaluate the integral: ft² ns te" du ds dt

A: Let's evaluate given integral.

Q: 2. Show that V(V × F) = 0, here vector field is twice differentiable. %3D

A: This question is from vectors.

Q: Consider the following integral 3dx V(16 + 4.x2 )5

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Q: Find the conservative vector field for the potential function by finding its gradient. f(x, у, 2)…

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Q: Show that the vector field F = x²i + (2yz)j+ y² k is conservative and then find the corresponding…

A: A vector field F is conservative if curl(F)=0 or ∇×F=0 The potential function is given by f such…

Q: 1,1,0 ex 0,0,0 Show that the vector field defined under the integrals e koct and that the scaler…

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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: (xy³ +3x²y, 3xy² + x²) 2+x ху у,

A: According to question given that Vector Field F⟨xy3+3x2y, 3xy2+x2⟩

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

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Q: Use the first Fundamental Theorem of Calculus and properties of integrals to explain why the…

A: Theorem used:First Fundamental Theorem of calculus:

Q: The work done by a force field in a moving particle around a closed path is zero.

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Q: In Exercises 5-40, evaluate the integral using FTC I.

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Q: Show that F of F. = (2xy + z³)i+x²+3xz²k is conservative force field. Find the scalar potential

A: *By Bartleby policy I have to solve only first one as these are all unrelated problems.These are all…

Q: Is the area F = 16xi+ (2 − y2) j conserved? If the conserved field is, find the potential function.

A: Here we find the potential function by integration.

Q: Find the outward flux of F = (4x +25y*,0,) 10z through the surface 25 4 9

A: Solution:-

Q: 6. What is the corresponding vector fieldV for the scalar potential p(x, y)= In(x+ ¥)?

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Q: Evaluate both sides for the field- rectangular parallelepiped X = 0 and 1 Y=0 and 2 and Z=0 and 3.…

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Q: Find the flux of the field F up through the part of the 1+ xy paraboloid z = 1 – x² – y? that…

A: FORMULA: Flux=∬F.n^ dS According to Stokes theorem, ∬F.n^ dS =∬∇×F . n^ dS According to the…

Q: Find potential 9, for the conservati ve field the vector F (x, y,a)- (2xe? - y*sinx) î * sinx) î +…

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Q: In 3 [te' i + e'j + In tk] dt 1

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Q: Use the local linear approximation of (1 – x)2º at xo O to estimate (0.99)2º. %3D

A: The linear approximation of f(x) at x0 is given by L(x) = f(x0) + f'(x0) (x-x0)

Q: Solve by Laplace transform Utt - 9urz = 0, 0 0 u(0, t) = u(T, t) = 0, %3D u, (x,0) = 0, u(x,0) = 2…

A: Use Laplace.

Q: Express the field D = (x2 + y)-(xax + yay) in cylindrical components

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Q: Find the work done by the force field F in moving an object from P(-4, 8) to Q(6, 4). F(x,y) = 2* 1…

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Q: Show that the vector field F(x, y, 2) = (2x cos y – 3)i – (x² sin y + 2²)j – (2yz – 2)kis a…

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Q: Evaluate the integra 7/t dt 7/t dt = t2 %3D

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Q: Use a parametrization to find the flux n do across the surface in the given direction.

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Q: he line integral of the vector field F= over the curve C which is the quarter of the rcle from…

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Q: Given the scalar field ∅= 2x^2yz^3 . Solve for the curl(∅ grad ∅)

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Q: A particle at (5, 0) traverses the upper semicircle x² + y² = 25 and then closes the loop along the…

A: Formula: The work done by the force F→ is given by, ∮CF· dr, where C is the path along the region.…

Q: find the volue of the od specify ingrol integrol T-x2 the

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Q: In Exercises 15-34, calculate the integral.

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Q: The integral of the vector field F(x, y) = (cos |x|, sin y|) A = (-2,0) and final point B (0, 2),…

A: Correct answer is (B) sin(2) - cos(2) + 1 Explanation:

Q: Find the work done by the force field F in moving an object from P(-9, 5) to Q(3, 5). F (r, y)

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Q: Exercise 1 Show that the integral domain Z[V10] = {a + b/10 | a,b € Z} is a FD.

A:

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.