Evaluate line integral of vector field F(x, y, z) = y sin(x²) i+ xyz j+ (x +y + z) k along line segment from the point (0,0,0) to point (1,2, 3).
Q: What is the line integral of a vector field?
A: To define the line integral of a vector field.
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Q: Find the gradient vector field of f(r, y) = In(x + 3y)
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Q: A particle moves along line segments from the origin to the points (1,0,0), (1, 2, 1), (0, 2, 1) and…
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Q: Find the conservative vector field for the potential function by finding its gradient. f(x, y) =…
A: It is required to find the conservative vector field for the potential function given in the…
Q: Find the flux through through the boundary of the rectangle 0 < x < 6,0 < y < 3| for fluid flowing…
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Q: A particle moves along line segments from the origin to the points (1, 0, 0), (1, 5, 1), (0, 5, 1),…
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Q: Use the Fundamental Theorem for Line Integrals to find the work done by the conservative vector…
A: If F(x,y,z) is a conservative field then there exist a scalar function f, such that ∇f = F. By…
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Q: Find the gradient vector field of f бу f(x, у, 2) = X COS Z (бху) (6y бу -sin бх -sin бу i cos k…
A: Given,
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Q: Calculate the outward flux of the vector field F→ (x,y,z) = 2xi→ + yi→ +zk→ along the boundary of…
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Q: Show that the line integral is independent of path. | 2xe Ydx + (2y - x²e)dy, Cis any path from (1,…
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Q: The directional derivative of the field u(x, y, z) = x² – 3yz in the direction of the - vector (i +…
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Q: A particle moves along line segments from the origin to the points (3, 0, 0), (3, 5, 1), (0, 5, 1),…
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A: To find the gradient field associated with the function φ(x, y, z) = xyz.
Q: 2的 在年
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Q: Find the gradient vector field (F(x, y, z)) of f(x, y, z) = In(x + 3y + 4z) . F(2, y, 2) =
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Q: Find the curl of the vector field F. F(x, Y, 2) = arcsin yi+ V1 - xzj + yêu y²k
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Q: Find the work done in moving a particle in a force field F = 3xyi - 5xyzj + 10xk along: straight…
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Q: Suppose a particle travels clockwise on a triangular path connecting the points (0,0), (3,1), and…
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A: The vector field Fx,y=fx,y,gx,y.
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A: we have to find the outward flux
Q: Find the conservative vector field for the potential function by finding its gradient. f(x, y, z) =…
A: Given: f(x, y, z) = 9x2 − xy − z2
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Q: Find the flux of the vector field F = across the part of the plane | z = 2 + x + 3y above the…
A: Given: F¯=y,-z,x z=2+x+3y The formula to determine the flux is: ∫S∫F·nds=∫D∫(-agx-bgy+c)dA Here…
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Q: Consider the conservative vector field 4 8 F (x, y, z) 3y cos (372), 37y sin (372) – a² > 2xz, 2x +…
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Q: Find the conservative vector field for the potential function by finding its gradient. f(x, y, z) =…
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Q: Find the gradient vector field of f(x, y) = x° sin(9y). F(x, y) = x°i+ sin(9y)j F(x, y) = 9x°…
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Q: Find the conservative vector field for the potential function by finding its gradient. f(x, y,…
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Q: Show that the vector field F = 2xyi +(x² - 2y)j is conservative and evaluate the line integral (F•dš…
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Q: Find the gradient vector field (F(r, y, z)) of f(x, y, z) = x² sin(3yz) F(z, y, 2) = (
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Q: Use Stokes' theorem to evaluate the work done by the force field F = ((f(x, y, z), 5xy, x+ yz),…
A: As per the question we have to find the work done, which is basically a line integral of the vector…
Q: Find the conservative vector field for the potential function by finding its gradient. h(x, y, z) =…
A: We have to find the conservative vector field.
Q: Show that the line integral is independent of path. 2xe-Ydx + (2y - x2e-Y)dy, Cis any path from (1,…
A: The product rule of differentiation states that d dxfg=fdgdx+gdfdx, where f, g are functions of x.…
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- find (a) div(f) (b) curl(f) if f=x^2(vector i)+y^2(vector j+2yz(vector k)Find a vector tangent to the curve of intersection of the two cyclinders x2+y2=32x2+y2=32 and y2+z2=32y2+z2=32 at the point (−4,−4,4)(−4,−4,4).A particle travels along curve C be described by vector function →r (t) =< 2t, 2t, 4 − t >. Find the distancethe particle travels from (2, 2, 3) to (4, 4, 2).
- Find a vector equation for the curve of intersectionof the surfaces x^2 + y^2 = 4 andz = xy.Find a vector function that represents the curve of intersection of the paraboloid z=7x^2+5y^2 and the cylinder y=5x^2. Use the variable t for the parameter.r(t)=⟨t, , ⟩Find a vector function that represents the curve of intersection of the surface 4x+2y-8z^2=16 and the cylinder of radius 3 wrapped around the y-axis.
- Vector v with initial point P = (x1, yı) and terminal point P, = (x2, y2) is equal to the vector v = ()i + (_j.The position of a particle is determined by the vector-valued function r(t)=<1-t^2, 3t, t^3>. Find the decomposition of the acceleration vector in terms of its tangential and normal components when the particle is at the point (0,3, 1).A hiker begins at the origin and walks in a straight line in the direction of the vector (1,1) for 2/2 kilometres, then in the direction of the vector (–1, 1) for /2 kilometres, and then in a straight line back to the origin. (a) Plot the path of the hiker in the xy-plane, clearly marking the values of the coordinate where there is a change of direction. (b) Calculate the total distance travelled by the hiker. (c) By regarding the xy-plane as contained in R³, use an appropriate formula involving cross products to compute the area enclosed by the path of the hiker.
- 4 Find the directional derivative of f (x,y)= 2x^2 y^2 -3xy at the point (1,-1) in thedirectiona. of vector v =<4,3> and toward the origin.Find the directional derivative of f(x,y,z)=yz+x^4 at the point (1,3,2) in the direction of a vector making an angle of 5π/6 with ∇f(1,3,2).Calculate the integration of (A" - dlP2P1 from point P1 (2,1, -1) to P2 (8,2, -1) of the vector function given as A° = ye"x + xe¯y on the line joining the two points.