Calculate the flux of the vector field F = x²i+ zk across the portion of the plane described as z = 4 – x - y for 0

Q: Find the flux of the vector field F = (y, z, x) across the - part of the plane z = 4+3x+2y above the…

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Q: Sketch the vector field F(x, y) = xj, the line segment from (1, 3) to (9, 3), and the line segment…

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Q: Use a computer or calculator with Euler's method to approximate the flow line through (1, 2) for the…

A: v→=y2i→+1.3x2j→ Initial point 1,2 ∆t=0.1

Q: .) Compute the flux of the vector field F = 4x + 4yj through the surface S, which is the part of the…

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Q: The figure shows the vector field F (x,y) = {2xy,x2 } and three curves that start at (1,2) and end…

A: Given, vector field F→x,y=2xy,x2 and three curves that start at 1,2 and end at 3,2.

Q: Find the flux of the vector field V(x, y, z) = 9xy²i + 9x²yj + z³k out of the unit sphere.

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Q: Compute the flux of the vector field (ï³, — xy³), out of the rectangle with vertices (0,0), (4,0),…

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Q: Consider the variable coefficient linear non-homogeneous ODE where and where u and are unknown…

A: The given problem is to find the values of u' and v' for the which the particular solution yp =…

Q: Find the flux of the vector field F = across the part of the plane z = 3 + 2x + 4y above the…

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Q: Find the gradient vector field of f(x, y, z) = x³y* + y° z³ + xz°.

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Q: Let C be the curve which is the union of two line segments, the first going from (0, 0) to (1, 4)…

A: As per Bartleby guidelines I have authority to answer only one question. If you want other…

Q: Let F and G be vector fields with differentiable components. Express curl (F × G) in term of div and…

A: Let F and G be vector fields with differentiable components.Let We will compute .The cross product…

Q: he position of a particle is determined by the vector-valued function r(t)=. Find the decomposition…

A: Given position of the particle We have to find the decomposition of the acceleration vector in terms…

Q: Compute the vector assigned to the pointP = (−3, 5) by the vectorfield:(a) F(x, y) = (xy,y − x)(b)…

A: To Determine : (a) F(x, y) = (xy,y − x)(b) F(x, y) = (4, 8)(c) F(x, y) = 3x+y , log2(x + y) Given:…

Q: a) Show that F (x, y) = (yexy + cos(x + y)) i + (xexy + cos(x + y) j is the gradient of some…

A: Given the function F(x,y) is the gradient of some function.

Q: Calculate the flux of the vector field F(x, y, z) = (4x+8) through a disk of radius 6 centered at…

A: Given,                    To find the flux of the given vector field through a disk of radius 6…

Q: Find the gradient vector field of f(r,y, z) = x*y* +y³z³ + æz°.

A: The gradient vector field of f(x,y,z) is given by ∇f=∂∂xf(x,y,z) i^+∂∂yf(x,y,z) j^+∂∂zf(x,y,z) k^…

Q: Find the parametrization of two different curves from the point (2,4) to (3,9). Compute the work…

A: Given point (2,4)  and (3,9)now equationof line passing through two point is given by…

Q: Let F = be a vector field in space Find the potential function f(x,y,z) of F

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Q: Find the flow of the vector field a = (x + 2y)i + (x + 2z)j − zk along the outer surface of the…

A: Given To find the flow of the vector field a→ = (x+2y)i→+ (x + 2z)j→ - zk→along the outer surface of…

Q: Compute the flux of the vector field (x², - xy³), out of the rectangle with vertices (0,0), (5,0),…

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Q: A constant field vector is given by E = -5a_p + 10a_0 + 3a_z. Calculate the vector component of E at…

A: The problem is related to vector. The topic is multidimensional geometry. We find the component of…

Q: Find the curl of the vector field u = a) b) ▼xu =( V. (V x u) = = (7xyz², 6x² y³, 2x³ y³ z4 ).

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Q: Compute the flux of the vector field F = yi + 5j - xzk through the surface S, which is the surfac y…

A: A vector field F→=yi^+5j^-xz through a surface S, which is the y=x2+z2  , x2+z2≤1. To Find: The flux…

Q: f(x,y) = 2x3+4y3-3x2-120x-192y+5 Is f(x,y) a curve, vector field, or surface?

A: To check whether it is a curve, vector or surface. f(x,y) = 2x3+4y3-3x2-120x-192y+5

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

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Q: If a field K is algebraically closed, then every polynomiel with co efficients in k has a root in K.…

A: A field K is said to be algebraically closed if every polynomial with coefficients in K has a root…

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: