Let F(x, y, z) = z* +x+y sin x +x In z 0, where z =f(x, y), then at (0,0,1) and %3D are equal to:

Q: The function f(z) =e² is: u(x.y)=e*cos(y) o u(x,y) =e*sin(y) ¤ contiuous in СО discontiuous in CO

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Q: Evaluate the line integral of f(x,y) along the curve C. f(x, y) = y + x, C: x2+ y2 = 1 in the first…

A: Line integral

Q: r cos(y) + y(e² - 1) x + 0 f (x, y) = ry x = 0.

A:

Q: Let /:R'R. S(x. y.2) = (e' tan y,x' In y). Compute D(f).

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Q: 6. Evaluate the triple integral f, x?yz dV, B = {(x, y, z): - 2 < x<1,0 < y< 3,1<z<5}

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Q: fyx if f(x, y) = sin(xy) + x² In y cos(xy)- xysin(xy) - cos(xy) + xysin(xy) + cos(xy) – xysin(xy) +…

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Q: f(x) = x² + sin z %3D is an antiderivaive of Athat satisfies F(x) = 0.| %3D

A: Given;- fx=x2+sin x   Solution:- ⇒∫x2+sin x dx ⇒∫x2 dx+∫sin x dx ⇒x33-cos x ⇒x33-cos x+C…

Q: Let f(x, y, z) = x'e sin(x yz) %3D 12 find dyəzd' x at the point (2, 0,0)

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Q: Let h(y) = In(y In y). Find h' (y). Show your work.

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Q: Where is the function f(x, y) = In Lütfen birini seçin: a. {(x, y) = R² | y-1} c. {(x, y) = R² y +…

A: Given that, fx,y=lnπ4+arctany+x ∴ log is defined when, ⇒π4+arctany+x&gt;0               1

Q: 5xy dv, where E lies under the plane z = 1 + x + y and above the region in the xy-plane bounded by…

A: The given integral is ∭E5xydV. The region E lies under the plane z=1+x+y and above the region in the…

Q: Calculate $o (3x + 5y - cos(y)) dx + x sin(y) dy if C encloses a region of area 4.

A: To evaluate the given integral.

Q: 9. Use Green's Theorem to evaluate (e* +y²) dx +(e" +x²) dy where C is the boundary of the region…

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Q: What is f'(x) given the integral function f(x) = cos(e“)d u? %3D O rm = 3x cos(e) O f'(x) = 3x cos O…

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Q: integral SS₁₁) dx dy xy=1, y = x, x=2 Calculate the in D limiteel to

A: An integral is a mathematical object that can be interpreted as an area or a generalization of an…

Q: 5. Let F (x, y) = (4xcos(y))ï – (2x² sin(y))j. (4xcos(y))i – (2x² sin(y))j. Find a function f such…

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Q: Find the double integral [[ f (x, y)dA for the function f(x, y)=1-6x²y and R:0<x<2,-1 <x< 1.

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Q: Define the map f : R² → R² as f(x, y) = (e" cos(y), æ²y + sin(y)) . (a) Show that f is…

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Q: f(z) =|z| is continuous in C 0 صواب ihi O

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Q: If f(x, y)is integrable function on the rectangle R = [a, b] x [c, d] where [a, b] on the x- axis…

A: In the question it is asked to show how the two double integrals are equal to each other.

Q: Let F= [y, z, x] and r(t)= [cos(t), sin(f), f] where 0sts n. Then the integral F(r0)dt is equal to…

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Q: 149. sin x cos ydx + (xy + cos x sin y)dy, where C is the boundary of the region lying between the…

A: We have to find ∫c sin x cos y dx +(xy+cos x siny)dy , where C is the boundary of the region lying…

Q: f (x, y) = xlx - alny is %3D O homogeneous 0 degree O not homogeneous O homogeneous 2nd degree O…

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Q: Find f, fy, and f. f (r, y, z) = z In (a*y® cos(2z)) f%3= fy = %3D fz =

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Q: Show that f is continuous at (0, 7) where a cos y + y(e" – 1) if r + 0 f(x, y) = ay if æ = 0.

A: In the given question we have to show that the given function f(x,y) is continuous at the point…

Q: The function f(z) =e² is: O contiuous in C discontiuous in C Ou(x,y) =e*cos(y) Ou(x,y)=e*sin(y)

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Q: If f(x,y)is integrable function on the rectangle R = [a, b] x [c,d] where (a, b] on the x- axis and…

A: True

Q: The power spectral density at zero frequency is equal to the area under the curve of the…

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Q: If f(x, y, z) = ln(æª + y!0 + z®) + V(x10 + y8 + zª), then find fr, fy and f; at (1, 1, 1).

A: Given, f(x,y,z)=ln(x4+y10+z8)+(x10+y8+z4)

Q: Compute the Wronskian of a set of functions {vı, y2} = {e* cos(/), e* sin(v¤)} cos (Væ), e* sin(va)}…

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Q: (a) Show that the limit xżycos y lim (x.y)→(0,0) 2x+ +y² does not exist. (b) If u = x cos y+ ye»,…

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Q: - Let f(z) = cosh z. (1) Find the image of S := {z = x+iye C: x < 0,0 < y < } under f.

A: Follow the steps.

Q: (a) Let Ty if +y +0, g(x, y) = 0 if r = y = 0. Prove that g(r, y) is continuous at (0,0).

A: To Prove: g(x,y) is continuous at (0,0)

Q: 2. Let z = g(x, y) = e" cos y + y In(x – 1). %D Use local linear approximation to estimate g(1.99,…

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Q: If f(x,y)is integrable function on the rectangle R [a, b] x [c, d] where [(a, b] on the x- axis and…

A: On rectangular region. By Fubini's theorem The given Statement is true.

Q: #2. Evaluate the triple integral x dV, E where E is given by E = {(x, y,z) : z< x < 2z, 0 < y<x+z,…

A: Solve it

Q: B// If w= f(x,v,z) is differentiable and x,y and z are differentiable functions of t, the w is a…

A: w=f(x,y,z) is a differentiable and x,y, and z are differntiable function of t.…

Q: Let U (x, y) = e-(y sin x Find a real-valued function V (x, y) such that f(z) = U + iV is analytic.…

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Q: Find a potential function for F(x, y, z) = (y² cos z, 2xy cos z, –ry² sin z). %3D COS

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Q: Let f(x,y) = x sin ry. Find: (a) fr(T,y) (b) fy(x, y)

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Q: Consider f(t) = (cos t, sin t, t²) and g(x,y,z) = (x² + y² + z², 3x + 4y –- 12z) and h = g(f(t)) =…

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Q: Let f (x, y) = if (x, y) # (0,0) and f (0,0) = 0. (z*+6y") a) Show that (0, 0) and (0,0) exist. dy…

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Q: Consider p(x, y, z) = 2xyz - x²y³ v(x, y, z) = xy² + y sin(z) F(x, y, z)= F₂(x, y, z)i + F₂(x, y,…

A: Let F→=f1, f2, f3. The gradient of the scalar field: g is defined as: ∇g=∂g∂x i^+∂g∂yj^+∂g∂zk^. The…

Q: 1 Integrate the function f (x, y) = over the region bounded by r-axis, line æ = 7, and the curve y =…

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Q: Integrate f(x, y)= y cos xy over the rectangular O ≤ x ≤ π, 0 ≤ y ≤ 1

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Q: Find the total differential of the following: 1. f(x, y) = sin x+ ye 2. f(x, y, 2) = – tan"(x+ y+ z)…

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Q: (a) Show that y = x* is concave upwards at the origin and y= e* is everywhere concave upwards.

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Q: Let f(r, y) = x sin ry. Fi (a) fz(x,y) (b) fy(r, y) Sy(x, y) (c) fyy(", y) YY (d) fyr(r, y)

A:

Q: b) Evaluate [[ (3-4x)dV where E is the region below :=4-xy and above the region in the xy -plane…

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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: