Determine if the vector field is conservative. 1 -) (yi + «j) F(x,y) = 1+ xy, conservative not conservative

Q: Determine the divergence of vector field E = 3x²x + 2zŷ + x²z2 at (2, –4, 0).

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Q: The gradient of a scalar field V is a vector that represents both magnitude and the direction of the…

A: Given : The gradient of a scaler field V is a vector that represents both magnitude and direction…

Q: A particle starts at the point (-1, 0), moves along the x-axis to (1, 0), and then along the…

A: Consider the provided information, The work done by force F in moving particle along the path C is…

Q: Show that the outward flux of a constant vector field across any closed surface to which the…

A: We have to Show that the outward flux of a constant vector field across any closed surface to which…

Q: What does it mean if the curl of a vector field is zero throughout a region?

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Q: Compute the work done by the vector field F = in moving a particle along the circle x + y = 1 from…

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Q: Show that the vector field F = ( -z, 0, x) is orthogonal to the position ----+ vector OP at each…

A: Given data: The vector field in the Cartesian co-ordinates is given by, F=-z i+x k.  Here, I and k…

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: 13 Is this vector field F(x, y)=(ycosx)i +(sin x+ y)j being conservative? Please show.

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Q: Show that F = (2xy + z³)ï + (x²)j+3xz²k is a conservative field. Find the scalar potential and the…

A: Given F = 2xy+z3 i ; x2j + 3xz2kThe objective is to show that given vector fieldis coonservative…

Q: Why does a two-dimensional vector field with zero divergence ona region have zero outward flux…

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Q: A smooth vector field F has div F(2, 5, 5) = 2. Estimate the flux of F out of a small sphere of…

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Q: Determine the vector field of F. F(x, y) 3D yi — хј

A: Let us make a table to draw the vector field as shown in the image below.

Q: Find the work done by the vector field F =( y+x 2, 2%y+z, 2yz + 2) %3D the Curve C ( all over it)…

A: The given vector field is F→=y2+xz,2xy+z2,2yz+x22 and a vector field F→ connects points 1,-1,-1 to…

Q: Sketch a few representative vectors of vector field F = ⟨0, 1⟩along the line y = 2.

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Q: Find the divergence of the vector field F→(x,y,z)=xyzi→−yzj→+xzk→at the point (8,3,2) .   Enter…

A: The given vector field is : Fx,y,z=xyzi-yzj+xzk To find the divergence at 8,3,2

Q: Sketch the vector field in the plane along with its horizonta and Vertical a representative…

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Q: Compute the flux of the vector field (x, - xy*), out of the rectangle with vertices (0,0), (5,0),…

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Q: Determine whether the line integrals of the vector fields along the oriented curves are positive or…

A: We have, the line integrals of the vector fields along the oriented curves as

Q: (1 point) Show that the vector field F-yz Ч + zz 1 J-zuz 2 k is conservative by finding a scalar…

A: To find the scalar potential function f, assume

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: Explain why a potential function for a conservative vector field is determined up to an additive…

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Q: Verify the Divergence Theorem in the calculation of the flux of the vector field: through the…

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Q: Give three equivalent properties of conservative vector fields.

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Q: Sketch a two-dimensional vector field that has zero divergenceeverywhere in the plane.

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Q: 1. Verify the Divergence Theorem in the calculation of the flux of the vector field: through the…

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Q: Find the divergence of the vector field and give clear explanations

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Q: Compute the flux of the vector field (x2, - xy*), out of the rectangle with vertices (0,0), (4,0),…

A: Given,

Q: Compute the flux of the vector field (x, (æ°, – ay*), out of the rectangle with vertices (0,0),…

A: To compute the flux of the vector field x3,-xy4, out of the rectanglewith vertices 0,0,2,0,2,4…

Q: What does it mean to say that a vector field is conservative?

A: What does it mean to say that a vector field is conservative?

Q: Find curl F for the vector field at the point (9, -9, 5). F(x,y,z) = x²zi - 2xzj + yzk curl F =

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Q: Describe the different methods available for evaluating the integral of a conservative vector field…

A: To Describe: Different methods available for evaluating the integral of a conservative vecctor field…

Q: Provide two examples of physical phenomena in which the divergence of a vector field is nonzero.

A: Q: What is Divergence? A:  It operates on vector field (having magnitude + direction both) , for…

Q: Compute the flux of the vector field F = xi + yj + zk through the surface S, which is a closed…

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Q: Let divF(x, y, z) = 4x³ + 4y3 +4z³, where F represents the flow of a vector field. Estimate the flux…

A: Given divF→x,y,z=4x3+4y3+4z3radius r=0.1 Centered at point 1,1,1

Q: Explain how to compute the divergence of the vector field F = ⟨ƒ, g, h⟩ .

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Q: When can we apply the Fundamental Theorem of Line Integral to compute work done by a vector field in…

A: Fundamental theorem is applicable for conservative field.

Q: Express (5x + 7y, 8x + 5y, 0) as the sum of a curl free vector field and a divergence free vector…

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Q: Therefore, div(curl G) = sin(y) – sin(y) + 8 = Since div(curl G) is different from 0, we conclude…

A: Given that:div(curl G)=siny-siny+8=

Q: 6. Vector fields V and W are defined by V = (2x – 3y + z, –3x – y + 4z, 4y + z) W = (2x – 4y – 5z,…

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Q: Sketch the vector field of F = (-y,0) in the xy-plane. %3D

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Q: Calculate the divergence and the curi of the following vector field at the point (1. 2. 1) Fix, y,…

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Q: Knowing that the vector field given by the image function is conservative, determine its potential…

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Q: A smooth vector field F has div F(1,4, 4) = 7. Estimate the flux of F out of a small sphere of…

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Q: Let divF(x, y, z) 4x3 + 4y3 + 4z³ , where F represents the flow of a vector field. Estimate the flux…

A: Given that a small sphere of radius 0.1, centered at the point  (1, 1, 1).And                  div F…

Q: If the vector field A = âx3x² + âyy – â̟5z³ is given, express à in cylindrical and spherical…

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Q: Calculate F=, around the triangle with vertices: (0,0,2), (2,0,0) and (0,2,0). Can you use Stoke's…

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Q: 6. Show that the system with vector field -x + sin(y) F(x, y) : -y+ cos(x), admits no periodic…

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Q: Determine whether of not the vector field is conservative. If it is conservative, find a function g…

A: The given vector field is: f=2xyz4,x2z4,4x2yz3 The field is conservative if curl F is 0.

Arc Length

Arc length can be thought of as the distance you would travel if you walked along the path of a curve. Arc length is used in a wide range of real applications. We might be interested in knowing how far a rocket travels if it is launched along a parabolic path. Alternatively, if a curve on a map represents a road, we might want to know how far we need to drive to get to our destination. The distance between two points along a curve is known as arc length. 

Line Integral

A line integral is one of the important topics that are discussed in the calculus syllabus. When we have a function that we want to integrate, and we evaluate the function alongside a curve, we define it as a line integral. Evaluation of a function along a curve is very important in mathematics. Usually, by a line integral, we compute the area of the function along the curve. This integral is also known as curvilinear, curve, or path integral in short. If line integrals are to be calculated in the complex plane, then the term contour integral can be used as well.     

Triple Integral

Examples: