Find a flux of the vector field ä(M)= (2z– y)i +6xj + yk through a part of the surface S: x - 4y+2z – 4 =0, created as a result of intersection of the plane S with coordinate planes.

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: Find the flux of the vector field V(x, y, z) = 3xy²i + 5x²yj + 2z³k out of the unit sphere. Flux =

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Q: Find the flux of the vector field = (y, −z, x) across the part of the plane z = 1 + 4x + 2y above…

A: Step 1:Step 2:Step 3: Step 4:

Q: The surface H is defined by (x² + y²)² + z = 1; z ≥ 0. The vector field A(r) is given by A = (x, y,…

A: The surface H  is defined by    .The vector field  is given by   .We have to sketch the surface H.

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

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Q: - Find the circulation of this ā (M) = (x+z)i + (x − y )×j + z×k vector field in the positive…

A: The given vector field is a→M=x+zi→+x-yj→+zk→. We have to find the circulation of the given vector…

Q: 3. Find a vector function that represents the curve of intersection of the paraboloid z =. and the…

A: To find the vector function that represents the curve of intersection of the paraboloid z=x2+y2, and…

Q: a) Calculate the flux of the vector field F(x, y, z) = 275k through a sphere of radlus 4 centered at…

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Q: calculate the flow of the vector field a(M) = (x-y)i + (x+y)j +(z2)k through the surface of a…

A: aM=x-yi^+x+yj^+z2k^ The equation of the cylinder x2+y2=1, z=0, z=2 The flow is in the direction of…

Q: 4. Consider the vector function r(z, y) = (x, y, x² + 2y*). (a) Re-write this vector function as…

A: For the given vector function, PART A The surface function is given by, f(x,y)=xi^+yj^+(x2+2y2)k^…

Q: Consider the vector field F(x,y,z) = (3x²y)i + (x³ + 1) + (92²) and the curve C made up of the…

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Q: (a) Calculate the flux of the vector field F(x, y, z) = 3i – 9k through a sphere of radius 2…

A: We will use divergence theorem to find flux through the sphere of radius 2 centered at origin .

Q: Find the flux of the vector field F = xi + e6xj + zk through the surface S given by that portion of…

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Q: 6 Find the constants a and b so that F = (ax'y°z°, x°y²z°, bx°y° z²) is conservative. You may use…

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Q: Using arrows, clearly sketch the vector field u(r) = (y², x + y, 0) in the (x, y)-plane.

A: Given the vector field function u(r)=(y2,x+y,0). We need to plot the vector field in the…

Q: Determine the vector field of F. F(x, y) = 81-4)

A: The given vector is F=8i-4j

Q: Find the flux of the constant vector field i = -4 i – 5j- 2 k through a square plate of area 16 in…

A: The solution is given below

Q: Let I be the flux of G = (4e³,9x³ex³,0) through the upper hemisphere S of the unit sphere. (a) Find…

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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: