(ii) Hence, use your answer obtained from Part b(i) to find the work done by the following field: F(x, y, z) = 3x²yi – 3zy²j + (2z – y³)k

Q: gral of the field F(x,y,z) = 7xi + 7yj + 7zk across the portion of the sphere x2 +y² +z? = a? in the…

A:

Q: 4. If div F >0 in a region enclosed by a small sphere, the outward flux of the field across the…

A: Given: Divergence F>0 is enclosed by a small sphere in a closed surface . The objective is to…

Q: verify the conclusion of Green’s Theorem by evaluatingboth sides of Equations (3) and (4) for the…

A:

Q: Find the wark done by the Foree Field FoK x>ye^> along the parabola X=y+1 from (bo) to (2,1).

A:

Q: 6. Let F = (x³ sin y, a* cos y). COS (a) Is F a conservative vector field? (b) If your answer to…

A:

Q: 43.) Show by checking cross partials that F = (4x³y² – 2xy³, 2x*y – 3x²y² + 4y³) is conservative Use…

A:

Q: For which value of a the field i = (y –x)i +(2x – y)j + azk is solenoidal. Find the gradient of the…

A:

Q: find the work done by each field along thepaths from (0, 0, 0) to (1, 1, 1) in Exercise 1.F = 2xy i…

A: Given: F=2xyi+j+x2k To Find: Find the work done by each field along the paths from 0,0,0 to 1,1,1.…

Q: 11. Show that the quotients field of the integral domain Z[i] = { a+ib: a, b e Z} is K={x+iy: x, ye…

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Q: Find a potential function f for the field F. F = (2y+z)i + (2x + 5z)j + (x + 5y)k A general…

A:

Q: Given a scalar field f(x, y)= xy + x² and C is the curve defined by the equation y =-x². Calculate F…

A: Given scalar field  fx,y=xy+x2 and C is the curve defined by the equation y=-x2.

Q: Q2: Find the work done S. F.dr in the force field F = yzî + xzĵ + xyk in moving from (1,1,1) to…

A: The equation of a line joining by the points and is given as the following.

Q: The work done in carrying a -1C charge from point A(0,0,0) to the point B(6,0,0) in the field E =…

A: Given,     E=2x.ax^+ 2yay^  VmQ=-1CPoints are A(0, 0, 0)  and  B(6, 0, 0)

Q: (a) Find a slope field whose integral curve through (1, 1) satisfies xy³ −x²y = 0 by differentiating…

A: (a) Consider slope field whose integral curve through 1,1 satisfies xy3-x2y=0 by differentiating…

Q: find the work done by each field along thepaths from (0, 0, 0) to (1, 1, 1) in Exercise 1.F = 2xy i…

A: Given- F=2xyi+x2j+k To Find- The work done along the path from (0,0,0,) to (1,1,1). Concept Used-…

Q: For Problems 2–3, you are given a vector field F and a curve C. Use the Fundamental Theorem of Line…

A:

Q: |Determine whe ther He Veetar Field Con ser vattive, iF it is Find Function For the veetor Field is…

A: Since, a vector field is defined by the function:-F⇀(x,y)=7yxi^ -x7y7j^ +0 k^ -(i)Our Aim is to…

Q: Find a potential function f for the field F. F = (5y + 2z)i + (5x + 3z)j + (2x + 3y)k A general…

A:

Q: Given: r = F| i = (x, y, 2) And letting F be the vector field F = f'/rP. Please solve in general the…

A:

Q: 2 find the divcrgence and curl of the vectur field f gven by F p) f(xyz) =cas (2x)it X sin ryit z°k…

A: the given vector field F is: F(x,y,z)=cos2xi+2sinπyj+z2k we have to find the divergence and the curl…

Q: Is the vector field F(x, y) = (sin(x)ely=3 + 2r cos(r>)y°, -2e)y-3 + 6 sin(x³)»*) conservative?…

A: Consider the given vector field.

Q: Q4// A// Evaluate both side of divergence theorem for the field D=2xy.x^+x².y^ C/m2 and the…

A:

Q: Find a potential function f for the field F = ev *2=(4 i + 4x j + 8x k). A general expression for…

A: We can calculate the following question by usual method of calculating potential function

Q: Find the work done by the field F(x,y, z) = (6z, 12x, y?) along the curve r(t) (cos(t), sin(t),…

A:

Q: Find a potential function f for the field F. F = (4y + 5z)i + (4x + 3z)j + (5x+3y)k A general…

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Q: 5. Show that F =(y cos(x),sin(x)+ y) is a conservative vector field and use this to calculate the…

A: Given: F → = y cos(x), sin(x) + y. A vector field F→ = P, Q is said to be conservative if partial…

Q: The slope field oY = f(t,y), with initial condition f(0) = 1, is shown. Find limf(t.y). dt - 10 「 ンン…

A: On the slope field draw the graphical solution with initial condition f(0)=1:

Q: Find the work done by the three-dimensional inverse-square field F(r) : 1 r on a particle that moves…

A:

Q: Is the following field path-independent/conservative? Cos(xy)i+ cos(ry) i F(x, y) 5xy 2 - No Yes

A: The given field is of the form Fx,y=Pi+Qj. Here P=5xy-cosxyx,Q=5x22-cosxyx.

Q: 4. Show, in detail, that any vector field of the form (*,J, 2) = J (*) +8(y)/+h(z)k is irrotational,…

A: To show that the vector field is irrotational.

Q: 6. Find the circulation of the field F(x, y) = -yi + x²j along the closed elliptical path x² + = 1…

A:

Q: Find a potential function f for the field F = 2y + 5z (4 i+ 8x j + 20x k). A general expression for…

A:

Q: Determine if F = (12r – 2x cos y, 2y + x² sin y) is a conservative vector field. If it is, find a…

A: Given that F=(12x3-2xcosy)i+(2y+x2siny)j. We need to determine whether if F is conservative field or…

Q: 1).Evaluate ∮CF•dr, whereF=⟨e^x−y^3,cosy+x^3⟩, and C is a circle of radius 2, centered at the…

A:

Q: The vector field A is given by A = 2r cos(r* + 3y")i + 6y cos(2² + 3y*)j. a) Show that A is…

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Q: Use the method of isoclines to sketch the slope field for y =x +y-1, then draw the solutions…

A: In this question, we sketch the direction field for the differential equation y'=y+x2-1 To draw the…

Q: Find a field F = M(x, y)i + N(x, y)j in the xy-plane with the property that at each point (x, y) ≠…

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Q: 6. Compute ſ, F-·dr where f(x, y) = (y² cos x, 2y sin r) and C is the straight line path starting at…

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Q: Find the divergence of the field F. yj - xk F= 1/2 +: 2 OA. 3/2 2 -X O B. 3/2 2 X -y OC. 3/2 O D. 0

A:

Q: 'ind the work done by the three-dimensional inverse-square field (r) = r on a particle that moves…

A: Given problem:-

Q: Let F(x) e R³be a smooth vector field. Note with AF ²F + R'. E R³. ΔΕ Show that curl(curl(F)) =…

A:

Q: 12. Find the flux of the field F(x, y, z) = 4xi + 4yj + 2k outward (away from z-axis) through the…

A: Flux of the given field is given by

Q: Find a potential function f for the field F. 7y j+ y² +z? 7z 21 F = (In x+ sec(10x + 10y)) i+ sec…

A: Given that F = (ln(x) + sec^2 (10x + 10y)) * i + (sec^2 (10x + 10y) + (7y)/(y ^ 2 + z ^ 2)) * j +…

Q: Q5: a) Show that the vector field F = (x² - y² + x)i - (2xy + y)j, is irrational.

A: Given vector field is F→=x2-y2+xi^-2xy+yj^. To show: Vector field F→=x2-y2+xi^-2xy+yj^ is…

Q: 7. Find the outward flux of the field F(x, y) = -yi + x²j across the closed counter-clockwise curve…

A: Consider the given field: F→(x,y)=-yi^+x2j^ and

Q: Verify the conclusion of Green's Theorem by evaluating both sides of each of the two forms of…

A: As we know that Green's Theorem turns a line integral into a double integral. It helps to simplify…

Q: 2 2 Find the gradient field of the function f(x.y.z) = In /x ?. + 4y +z The gradient field is Vf =…

A:

Q: 1. For the scalar field 0 = x3 + 2xy in the domain ABC below, calculate the gradient field and then…

A:

Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.