a) Find the values of a,b,c which make f(x,y,z) = sin (xªy°z©) a potential function for the vector field F(x,y,z)= 2 x'y5z4 cos (x²ysz*) i+ 5x²y+z+ cos (x²ysz*) j + 4 x²y5z³ cos (x²y%z*) k a= b= C=

Q: A particle with mass 5kg is moving along the x-axis under the action of position and time dependent…

A: Find the answer below.

Q: Let 3 = (X'X)-¹X'Y as shown in class. Let y = w'ß, where w is a column vector. Show that Var(y) =…

A:

Q: (d) Let f(x, y) = 2x² + xy-y². Prove that the directional derivative of f(x, y) at point x = (3,-2)…

A:

Q: Find V· (V × F), if F(x, y, z) = 3e**i+3xe'j – 3e²k. - V·(V × F)

A:

Q: 4. Compute the first fundamental form.

A: Given xu,v=u2-cosv,u+sint, cosv

Q: af dy of f(x, y) = tan (x²y³ + 2xy)? What is the

A: Given:     To determine:

Q: Ax, y) = sin(xy), r(t) = (e²t, e³), t = 0 %3D

A: Solving by differentiation we have

Q: Given the vector field v=⟨0,2xz+3y^2,4yz^2 ⟩. Find the line integral of the path from (0,0) to…

A:

Q: (b) Verify your answer in (a) using Stokes' theorem, where the normal vector to the surface has a…

A: We have to apply Stoke's theorem.

Q: b) Show the force field (x, y, z) = (x²y + In(yz)) i+ + y cos(2z) ) j+ y? sin(2z) 3 F k is a…

A:

Q: of f. Sketch the gradient field for f (x, y) = /x² + y? along with several level curves

A: Here we have,

Q: Find domain and vange f (2y)- し、 (x'+ゴ) ) fCz, y) = Sin" (y-z) Ž (3,y)- tan

A: According to the question,  We need to specify the domain and range of the given function: (A) f(x,…

Q: Sketch the slope field and solution curves   a) yy' + 4x = 0, (1,1), (0,2)

A: The given equation is yy'+4x=0 for 1, 1 and 0, 2. To find: sketch the slope field and solution…

Q: le can define an operator * : 2'(R?) n'(R?) by defining *dx = dy and *dy = -dx, and extending…

A:

Q: 2. Solve V'u = 0 for r<1, 0<z< T, u(r, 0, 0) = 0, a7(r, 0, 7) = 0, - u(1, 0, z) = z cos³ 0.

A: It is given that

Q: 1 1 1 1 ++* 1 2 1 1 -2 1 1 Which system of ODEs matches the velocity field above? Ox'= 1, y = x…

A: The velocity field is shown in the picture. Consider, x'=yy'=1 This means: y'x'=1yydy=dx Integrate…

Q: Calculate Vf, V•f, V×f f (x,y. =)= xi+ y'j-* by the function

A: Introduction: The curl of the vector f(x,y,z)=fxi+fyj+fzk is given by ∇×f=∂∂xi+∂∂yj+∂∂zk×fxi+fyj+fzk…

Q: Use the Cauchy-Riemann Equations to show that f'(z) does not exist at any point. a) f(z) = z…

A: Solution :-

Q: EXERCISE 2 Consider f(x, y) = 3x² – 4y² and = = cos T î + sin 7 ĵ. Use the Theorem above to find the…

A: This is a problem related to vector calculus. Based on the general formula and concept we will…

Q: find the work done by the force field F(x,y) =(2y+x²,x²-2x) acting on an object as it moves along…

A: Given: The force field, Fx,y=2y+x2,x2-2x.   To find: The work done by the given force field acting…

Q: The magnetic field B due to a small current loop (which is placed at the origin) is called a…

A: Let S be a surface enclosed by the curve C , then ∬ScurlF→⋅dS→=∫CF→⋅dc→

Q: 5. Compute the lenght of the curve u = t, v=0 from t = 0 to t = 1 using the first fundamental form.

A: Suppose, a continuous function is provided over a certain interval. The arc length is the length…

Q: Let F = (t + 4) i +² sin(41) j+ t^k, G = e' i+ 4f* cos(t) j+ e$t* k and 512 -i+ 4² j+te²ª k 1 +t H =…

A: Given :  F=(t+4)i+t^ 2 sin(4t)j+t^ 4 k G=e^ t i+4t^ 4 cos(t)j+e^ 5t^ 2  H = (t ^ 5)/(1 + t) * i + 4t…

Q: a) b) O c) d) O 2 пу z = f(x, y) = cos (x² − 2x + 2) + x + 2) Which one of the following is the…

A:

Q: F(x,y,z) = ⟨1,y,z^4/3⟩ a. Find the parametric equation of a curve passing through the point…

A: Consider the given vector field. F(x,y,z)=1,y,z43 and  a curve passing through the point ro=-1,1,1…

Q: = (u² - cos v, u + sint, cos v) x(u, v):

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: