|| (r+ y)e° dA , where R is the rectangle enclosed by the lines r – y = 0, R I-y = 2, r+ y = 0 and r+ y = 3. Transform R into uv plane by making an appropriate change of variables in order to find the integral. (a) Plot R in both ay and uv planes. (b) Find the Jacobian of the transformation (c) Evaluate the integral using the transformation

Q: = (1 – t)i + (4 + t)j ess your answer as an ir erical answer only.

A:

Q: Sketch a diagram of the linear map G that maps the rectangle R = [0, 1]×[0, 1] in the uv-plane to…

A: Refer to the figure shown below:

Q: Sketch a surface representing a function f of two variables x and y. Use the sketch to give…

A:

Q: Sketch the level curves f(r,y) = c of the function f(x,y) = ln(r² + y²) for c = -1,c = 0,c = 1,c =…

A:

Q: Find the curl, the divergence and the potential (if it exists) of F = (2x + y², 2ry + 3y²z*, 4y°z³)

A:

Q: Find the Laplace transformation of the unit impulse function 8(t-22017) and The unit step function…

A:

Q: h) Find a parametric representation of the Parabolic equation z = 9(x² + y²)

A: Parametric representation of the Parabolic equation z = 9(x² + y²) will be cartesian coordinates as…

Q: If the covariant tensor has components in Cartesian coordinates x',x². Find its components in polar…

A:

Q: Consider the function F(x, y, z)=(√√1 - x² - y², ln(e² - z2²)). (a) This function is a mapping from…

A:

Q: Fourier's Law of heat transfer (or heat conduction) states that the heat flow vector F at a point is…

A:

Q: (15) Given two functions u(x + y) and v(x + y), what is Vu × Vv? Don't show it using an example. If…

A:

Q: Evaluate fS (3x + 4y²)dA by changing it into polar coordinates over the region in upper half plane…

A: In polar coordinates

Q: Q1J Use the Chain rule to find the derivative of w=xy + In z with re spect to v if x v/u, y u+v, z…

A: The chain rule is used to calculate the derivatives of the function. The chain rule for a function…

Q: Example Let f(r,y, z) = ryz sin rz. Find Vfo at the point Po = (1, –2, 7) and compute the…

A: to find : (1) grad f. (2) directional derivative.

Q: b) Find v²f where f = e²x sin2y. cl Write the sufficient condition for ovistence of Lapla

A:

Q: xy? calculate the dot product (x, y) ▼ : 2 + y Consider the f unction

A:

Q: Find the first and second fundamental form of x(u,v)=(e(u-v)/2 cos(u+v)/2, e(u-v)/2…

A:

Q: Suppose F 2.xz + 3y²) ay + (4yz²) az. (a) Calculate fLL F dS, where S is the shaded surface in…

A:

Q: Let FMN be the transformation of the surface. Prove F*dg=d (F*g) gEA (N) = S(N). where

A:

Q: et F = t+3 2i-fj+3t°k and G = e'i – 3f j+ e²* & %3D

A: Given are two vector functions F=t+3ti-t2j+3t3k and G=eti-3t3j+e2tk.

Q: a) Find the Laplace transformation of the function f(t) = cosh at sinh bt

A:

Q: V × (V × F) = V(V-F) – V°F. %3D

A:

Q: g) Find the Fourier sine series of the function f(x)=-100¹0(-<x<n); f(x)= 100¹0 (0<x<n)

A: We have to find the Fourier sine series of the function  f(x)=-10010if -π&lt;x&lt;010010if…

Q: Use Green's Theorem in the form of this equation to prove Green's first identity, where D and C…

A: ∬Df ∇2g dA=∮Cf(∇g).n ds -∬D∇f.∇g dA Add ∬D∇f.∇g dA to both sides, ∬Df ∇2g dA+∬D∇f.∇g dA=∮Cf(∇g).n ds…

Q: Fourier's Law of heat transfer (or heat conduction) states that the heat flow vector F at a point is…

A: Let F be a vector valued function over the surface S , the surface integral is given by   where V…

Q: Take a defineal a transformation. by OIYE Vectors in the w(x)- cCKed where c and d EU clidean plane…

A: In euclidean plane, d(x,y) = |x-y| ∀x,y∈ℝ.

Q: Apply Stoke's theorem to compute ||curl F-ds where C is the triangular closed path Toining the point…

A:

Q: Let f = (r +2ryz)i+( + 2xy)j+ (+ 2xyz)k. Find j at (1.0, 1). (a) divergence of

A:

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: