Find the flow of the velocity field F = (y + z)i + xj - yk, where velocity is measured in meters per second, over the curve r(t) = eti - e2tj + e-tk, 0<= t<= ln 2.

Q: Let y = V7 – x. %3D Find the differential dy when a = 4 and dx 0.3 Find the differential dy when x =…

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Q: Consider the slope field shown. (a) For the solution that satisfies y(0) = 0, sketch the solution…

A: Given, the slope field:

Q: Calculate the total differential dz for z = r sinh(ry) + yelv

A: Note- since you have asked multiple questions, So Acc. to Bartleby guidelines we are supposed to do…

Q: 8. Find the flow of the velocity field F(x, y) = ( + y)î – (x² + y² )j along the circle a? + y² = 1…

A: According to the given information, it is required to find the flow of velocity field.

Q: Let x and y be differentiable functions of t, and lets = √(x2 + y2) be the distance between the…

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Q: Use the total differential dz to approximate the change in z= 6-x² – y² as (x, y)moves from the…

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Q: Determine the total differential (dg), and total derivative with respect to x 3x 2y g f(x, y, z) 3D…

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Q: dz Z 1. around unit circle of radius 1 and center at (0, 0) counterclockwise. 2. By using direct…

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Q: Find the line integral of ƒ(x, y) = yex2 along the curve r(t) = 4t i - 3t j, -1 <_ t <_ 2.

A: Given: f(x,y)=yex2 r(t)=4ti-3tj -1≤t≤2 From r(t)=4ti-3tj x=4t,y=-3t Differentiate with respect to…

Q: If F= represents the flow of a liquid through a parabolic cylinder S: y=x, 0sxs2, 0szsl. That is…

A: If F→=y2,-x2,z2 represents the flow of a liquid through a parabolic cylinder S:y=x2,0≤x≤2,0≤z≤1.…

Q: 2. Show that f(t) =1 – sint defines a smooth flow on the manifold S = ( (cost,sint) ), but %3D g(t)…

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Q: Find the total differential of W = x^4 y^3 z^2 + x^3 Y^2 + x^2 z +y + 3 at (2,1,1)

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Q: (b) Find the derivative of g in the direction of a=2i+curl w at point (0,-1,1) where u u +2 g= In. J…

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Q: Calculate the work done by a constant force field F on a particle that moves once uniformly along…

A: Given, path of the curve x^2+y^2=1 &amp; F(x,y)=(αx,αy)

Q: Find the total differential. z = x cos(y) – y cos(x) dz =

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Q: Consider the slope field shown. (a) For the solution that satisfies y(0) = 0, sketch the solution…

A: Given,  the slope field :

Q: Find the total differential of the function. 4y f(x, y) 5x - y df =|

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Q: Evaluate the line integral curve C: x = = 4 cost, y (5x + 7y)dx + (3x − 6y)dy along the = 8 sint (0…

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Q: Use the differential dz to approximate the change in z = √4 − x2 − y2 as (x, y) moves from the point…

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Q: Find the total differential of the function f(x, y, z) = 4 ryz2 df ? dx + ? dy + ? dz

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Q: Find the total differential of the function f(r, y) = (3 y²+6 2):

A: Given,           The function is fx,y=3y2+6x13 .

Q: Find the work done by F(x, y) = (x, Y) over the curve r(t) = (e' cost, e' sin t) (x² + y?)3/2 from…

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Q: Use the Chain Rule to find dz/dt. dz dt || z = √√√ 1 + x² + y², x = 2 In(t), y = cos(t) 1 √1+ (Int)²…

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Q: Sketch the slope field for dy/dt = y sin t for −2π ≤ t ≤ 2π, −2 ≤ y ≤ 2.

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Q: Show that the displacement u(x, t) in an elastic string of length t fireed at both ends which is set…

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Q: Use the Chain Rule to find dw/dt. In(Vx² + y² + z² ), x = 3 sin(t), y = 5 cos(t), z = 6 tan(t) W =…

A: We have to find the derivative  using chain rule.

Q: Write the differential dw in terms of the differentials of the independent variables. w f(x,y,z)-…

A: To calculate the total differential dw for the given function of three independent variables

Q: Write the approximate change formula for a function z = ƒ(x, y) at the point (x, y) in terms of…

A: Consider the following function: z=fx,y

Q: Find the potential function for F(x,y,z) = (x^-1z)i + (y^-1z)j + ln(xy)k. Show all steps and explain

A: Potential function of  F, is f such that ∇f=F.

Q: Consider h(x, Y, z) = cos (xy) + eY% + ln (xz). Determine the directional derivative of h at the…

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Q: Sketch the slope field for dy dt = t y for −2 ≤ t ≤ 2, −2 ≤ y ≤ 2.

A: Sketch the slope field:  dydt=ty for -2≤t≤2, -2≤y≤2. Construct the table of Values. t y dydt 0…

Q: Use the Chain Rule to find dw/dt. w = In Vx2 + y2 + z2 ), x = 4 sin(t), y = 6 cos(t), z = 5 tan(t)…

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Q: Evaluate the line integral ∫(AB) xdx+ydy+zdz using the fact that the integrand is a perfect…

A: Here we find ∫(AB) xdx+ydy+zdz,   by using the fact that the integrand is a perfect differential. A…

Q: Use the multivariable chain rule to find du/dt at t = 1, where w = sin(r²y), 1 = Int, y = 3t + 1.

A: To Determine: use the multivariable chain rule to find dwdt at t= 1 Given: we have w=sinx2y…

Q: Find the differential of the function w = x* sin(y z' ) dw dx+ dy+ dz %3|

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Q: Verify the condition of integrability for the given total differential equatior (3x*y2 – e*z)dx +…

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Q: a)Let z = x?y – y² where x = t2 and y = 2t . Find dz/dt at t=1 using chain rule.

A: Given query is to find derivative is nge chain rule.

Q: Write the differential dw in terms of the differentials of the independent variables. w =…

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Q: Assume that a body moving with velocity v encounters resistance of the form dv/dt = -kv3/2. Show…

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Q: Consider F(x, y) = (y², 2xy + 1) . Let c be the curve defined by F(t) = (t³, t), for 0 <t < 2.…

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Q: Find dw/ds and dw/at using the appropriate Chain Rule. Function Values w = sin(4x + 5y) x = s + t, y…

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Q: Find z = f(x, y) and use the total differential to approximate the quantity. (2.04)^2(1 + 8.8)^3 −…

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Q: dy/dt = (y+1/2)(y+t) Use a slope field to explain why the solution that passes through (0,1) is…

A: A slope field is used to determine the approximate slope of the tangent at a point to the curve…

Q: Using an appropriate linearization, the local linear approximation of f(z)= T+ sin z at 27 is

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Q: Find the path of a heat-seeking particle placed at point P on a metal plate whose temperature at (x,…

A: Consider the given function, Tx,y=100-x2-2y2 the differentiate partially, ∇T=-2x, -4y the slope at…

Q: Q2) A) Verify that wxy = Wyx for w = x sin y + y sin x+ xy. B) Show that f(x, y) = e-y cos 2x…

A: Here we have to find the partial derivative of the functions with respect to x and y seperately…

Q: Find the total differential of w ze(z+y) at (0,0, 1).

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Q: (b) Calculate the rate of change of the function f(r, y, 2) = ryz along the path r(t) = (sin(t), e“…

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Q: Find the total differential. w = e7Y cos(4x) + z7 dw

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Q: 2. Show that f(t) = 1 – sint defines a smooth flow on the manifold S = ( (cost, sint) ), but g(t) =t…

A: The given question can be solved as shown in step2

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: