For the scalar function f, div(grad f) is equal to Select one: а. dy? -j+ dz? b. 0 C. afit -j+ dx ду dz d. Non of them е. ax? ay?' az?

Q: Determine whether the, vectur feild f(y)= r"cosy i- siny is Conservatvc. If it is, find a potential…

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Q: Find the gradient vector field of f(x, y, z) = 9x5 + 7y3 + 6z?

A: To find the gradient of  f(x,y,z)=9x⁵+7y³+6z²

Q: Find the gradient of f at the indicated point. f(x, y) = 3x² + 3y^; (4, 1) NOTE: Give your answer in…

A: We know that; The expression for the gradient vector: ∇fx,y=i∂∂x+j∂∂yfx,y=i∂fx,y∂x+j∂fx,y∂y Given:…

Q: In Exercises 31-36, find a parametrization of the tangent line at the point indicated. t,4t 2,2t 3),…

A: Differentiation: The line must pass through the point at t=-2 is  (t,4t+2,2t+3)=(-2,-6,-1)

Q: For y'= Aj where A is given below %3D A= find the right hand side vector F(t, ) needed to apply…

A: Given y'→=Ay→ Here A=1245 We have to find right hand side vector F→(t,y)→.

Q: Which of these is the correct parametrization of the line connecting (1,2,5) and (0,3,- 1)? A. x =…

A: A line connecting the points x1,y1,z1 and x2,y2,z2 can be calculated as,…

Q: Express the following as a vector valued function 7 (t). 1. The line L defined by y+3=3x and z+1 =0…

A: We will express the line L defined by y+3=3x and z+1=0 hyperbola -x2-2x+2y2+4y=-2 as a vector valued…

Q: Find a vector function that represents the cur ve of intersection of the paraboloid z a + mnd the…

A: Given that, the paraboloid is z=x2+y2 and the cylinder is x2+y2=22. The projection of the of the…

Q: What is the correct piecewise parametrization of a path C created by starting from C, of equation y…

A: Given a piecewise smooth curve C. The question is to parametrize the curve.

Q: If a vector function v(t) = (t, Int,t-²) is given for a scalar function Ø = x In z + Jy - cos(x + 1)…

A: Given the scalar function φ=xlnz+y-cosx+1. This scalar field is to be evaluated over the path…

Q: In Exercises 31-36, find a parametrization of the tangent line at the point indicated. 31. r(t) =…

A: The parameterization of the tangent line is given as L(t)=r(t)+t r'(t)

Q: State what it means for a scalar function y = f (x) to beintegrable on an interval [a, b].

A: Scalar integrable  function y = f (x)

Q: Consider the paraboloid defined by x-4y2+4z2. (a) Which of the following vector-valued functions…

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Q: Find a vector parametrization 7 (t) for the intersection between the two surfaces: x² + 10y4 + 223 =…

A: According to the given information, it is required to find a vector parametrization for the…

Q: For the scalar function f, div(grad f) is equal to Select one: а. a²f -k dz? i+ dx? b. Non of them…

A:

Q: For scalar functions u and v, show that B= (Vu) × (Vv) is solenoidal and that 1 . (uVv - υVu) 2 A is…

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Q: he upward flux of

A: Introduction: Let F be a function and let the region is S, then the flux of the function is given…

Q: Q1 show thot the force F-ljz-6xz+2xy2) ij+(3xyz-6x²z)is cors.ervotive, hence find the Potenial fum…

A: According to our guidelines, we can answer only the first question or first three subparts of a…

Q: Find the gradient vector field Vf of f. 1 f(r, y) = (y – 2)?

A: For a given vector fx, y the gradient vector denoted by ∇f is ∂ ∂ xf∂ ∂ yfSo,=∂ ∂ x12y-x2∂ ∂…

Q: In Exercises 31-36, find a parametrization of the tangent line at the point indicated. ( 1 - t 2 , 5…

A: Given: The line given is r(t)=1-t2,5t,2t3. Substitute 2 for t in r(t)=1-t2,5t,2t3.…

Q: Find a vector parametrization of the circle of radius 5 in the xy-plane, centered at the origin,…

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Q: Q6/ A boxful has been moved a distance (2m) in a F = 5.2 i+ 3J horizontal direction due to effect of…

A: Work done by a force is equal to the product of Force applied in the direction of distance and…

Q: Q1:- Find the domain of the following vector fumctions:- (a) f(t) = (cos nt)i - Ln(t)j + Vt-2k (b)…

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Q: Determine if the vector filed is conservative and if so find the potential function F (x, y) = 12xyi…

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Q: Find the gradient of following scalar function: |5x+7y'•3z r = V

A: Gradient of any function r=f(x,y,z) is given by the formula ∇r=∇f(x,y,z)=∂f∂x,∂f∂y,∂f∂z Where…

Q: Express the given parametrization in the form y = f(x) by eliminating the parameter. x = 5t + 2, y =…

A: Given

Q: Find the work done by the force F =xyi + (y - x)j over the straight line from (- 2, - 1) to (-1,0).

A:

Q: Given a differentiable vector function r(t) defined on[a, b], explain why the integral ∫ba||r'(t)||…

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Q: Find a parametrization of the tangent line at the point indicated. r(t) = (1 - t4, 4t, 5t), t = 2…

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Q: Find a formula for the vector field (vectors are shown as line segments, with a dot at the tail of…

A: From the given information.   Observe the given vector field.

Q: Find the gradient of f at the indicated point. f(x, y) = 3x² + yt; (4,1) NOTE: Give your answer in…

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Q: Compute the directional derivative of f(r. y) = = at the point (1, 1) in a irection normmal to the…

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Q: Find a vector parametrization of the circle of radius 5 in the xy-plane, centered at the origin,…

A: Solution: Given, Circle having center at (0, 0) and radius is 5 and

Q: Find the work done by the force F= xyi + (y - x)j over the straight line from (1,2) to (3,6).

A: we have to find the work done by the given force

Q: For the scalar function f, div(grad f) is equal to Select one: a. + + dx?' ay? ' az? b. dx dy dz C.…

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Q: Find the net outward flux of F =axr across any smooth closed surface in R, where a is a constant…

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Q: Let A = LU, where L is lower triangular with 1’son the diagonal and U is upper triangular. How many…

A: Given: Let A = LU, where L is lower triangular with 1’s on the diagonal and U is upper triangular.…

Q: Give the two unit vectors u such that the function z = g(x, y) of example 3 has zero derivatives at…

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Q: Why is r(t) = ⟨ƒ(t), g(t), h(t)⟩ called a vector-valued function?

A: Given r(t) = ⟨ƒ(t), g(t), h(t)⟩. We need to give reason why r(t) is called vector-valued function.

Q: Find the work done when a constant force F=16l bs moves a chair from x =2.4 to x =3.1 ft. along the…

A: Solve the following

Q: 4. If ricost cosh t+j sint sinh t, where i and j are constant orthogonal vectors prove that (a)…

A: Given r = i cost cosht + j sint sinht

Q: Find the work done by the force F (x; y) = xyi-(y-x)j over the straight line from (1,1) to (2,3)

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Q: Find the work done by the force F= xyi + (y - x)j over the straight line from (-1,- 2) to (2,1).

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Q: Q1:- Find the domain of the following vector functions:- (a) f (t) = (cos nt)i – Ln(t)j + vt – 2k…

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Q: The unit tangent vector of the function ⟨2t+1,(3t+1)^2,(2t+1)^3⟩ at the point with parameter value…

A: Topic - vectors Option d

Q: What is the gradient of the following scalar function, in spherical coordinates: singsinф f r

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Q: For scalar functions u and v, show that B = (Vu) × (Vv) is solenoidal and that 1 A = (uVv – vVu) is…

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Q: Find a vector parametrization of the circle of radius 5 in the xy-plane, centered at the origin,…

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Q: Find the natural parametrization of r(t) = 1²0₂ + 20y. What is this curve?

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Optimization

Optimization comes from the same root as "optimal". "Optimal" means the highest. When you do the optimization process, that is when you are "making it best" to maximize everything and to achieve optimal results, a set of parameters is the base for the selection of the best element for a given system.

Integration

Integration means to sum the things. In mathematics, it is the branch of Calculus which is used to find the area under the curve. The operation subtraction is the inverse of addition, division is the inverse of multiplication. In the same way, integration and differentiation are inverse operators. Differential equations give a relation between a function and its derivative.

Application of Integration

In mathematics, the process of integration is used to compute complex area related problems. With the application of integration, solving area related problems, whether they are a curve, or a curve between lines, can be done easily.

Volume

In mathematics, we describe the term volume as a quantity that can express the total space that an object occupies at any point in time. Usually, volumes can only be calculated for 3-dimensional objects. By 3-dimensional or 3D objects, we mean objects that have length, breadth, and height (or depth).

Area

Area refers to the amount of space a figure encloses and the number of square units that cover a shape. It is two-dimensional and is measured in square units.