3. For what value of K the velocity field in plane polar coordinates given by v, = tion of continuity? K (- 1) cos 0 and ve = K ( +1) sin 0 satisfies the equa-

Q: 1. Which of the following(s) is/are applicable to solve | sin(2y)dr – y cos(r)dy over an arc from…

A: To solve the given integral, we can use the following methods:

Q: 1. Use Green's theorem in order to compute the line integral $ (3cos + 6y?) dx + (sin(5") + 16x³) dy…

A: Here the given integral is                               ∮C3cosx+6y2dx + sin5y+16x3dy

Q: Find the length L of the curve R(t) = e* cos(3t) i + e* sin(3t) j + 3e*k over the interval [1, 5]. L…

A: given: Rt=etcos3ti+etsin3tj+3etk interval 1,5

Q: Find a parametrization of the right branch (x > 0) of the hyperbola(x/a)2−*(y/b)2= 1 using cosh…

A: Given that. xa2-yb2=1     ...(1) Need to evaluate parametrize form of (1).

Q: Show that the volume of solid formed by revolving one loop of the curve r2 = a? cos20 about the line…

A:

Q: 92 that the elect ric potential prove -5x V = e Cos 13y sinh 12 Z achieves the Laplace equation اثبت…

A:

Q: Which of the following integrals correctly solve for the magnetic flux passing through the surface…

A: We can solve this using magnetic flux formula

Q: (;)* + (4)° = 1 in the first quadrant, find the following line integral with respect to arc length.…

A: Find the line integral ∫C5x-3y ds where the curve C is the part of the equation x42+y42=1 in the…

Q: 3. Two fundamental differential postulates of electrostatics are given as: V•E = P a. where Р is…

A: The general form of a vector-valued function in three dimensions is given by E→=E1i+E2j+E3k, where…

Q: Assuming that C is the counterclockwise oriented boundary curve of the unit square in the first…

A: It is an simple and easy question on vector calculus  Can be easily solve by use of green's theorem

Q: Which among the following is NOT a LINEAR first order differential equ O (1+ cos 0) de sin 8 (sin 6…

A: Given that: 1. Linear first order differential equaiton.                  2. dydx+ln(x+1)yx=x3yk

Q: Is the differential 2re sin ydr+e cos ydy exact? If yes, then the funetion f from which it is exact…

A:

Q: A particle is moving in space with position function R(t) = (cos(4t), sin(4t),5 - 3t) %3D Find the…

A:

Q: Show that the spherical harmonics 1 3 sin 0 e-iº, 2V 27 Y1,+1 and 3 1 Y1,0 2V V cos 0, %3D | are…

A: Given: The given harmonics are Y1,+1=1232πsinθ e-iϕ and Y1,0=123πcosθ  are orthogonal over the…

Q: Consider the following. sin (18.35)* + (7.657) – sin(s² - 8°) Find z = f(x, y). f(x, y) = Use the…

A:

Q: Consider a particle moving along a path C given by r(t) = v3ti+ = V3ti sin (t)j + cos (t)k Find the…

A:

Q: Calculate the volue of the de integral 23 - 1 2 F soro fallowing c a) Cil2+1|= 1 Curves. 6)…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: Q1- The divergence of the function f-(rs cos 0)e, in spherical coordinates is......

A:

Q: Find a parametrization of the tangent line at the point indicated. r(t) =〈cos 2t,sin 3t〉, t = π/4

A: Given :  r(t) =〈cos 2t,sin 3t〉, t = π/4

Q: Find the length L of the curve R(t) = e" cos(2t) i + e* sin(2t) j+2e"k over the interval [1, 4]. L =

A:

Q: Consider the parametric curve: x = ln(t + 5) y = −6t5 Which integral gives the arc length of the…

A:

Q: What is the equation of the slant asymptote for the slope field created by the differential equation…

A: The objective is to find the equation for slant asymptote for the slope field created by…

Q: Evaluate the line integral (cos x + sin y) ds; C is the line segment from (0, 0) to (x, 67).

A:

Q: (10) Find the work done by F = -yi+ xj+ zk on a particle moving along the helix r(t) = where 0 < t <…

A:

Q: 2. What is the arc length of the polar curve r = 1- sin 0 for 0 <0< <5? 2 (a) 2(v2 – 1) 2 - V2 2/2…

A:

Q: (a) Suppose that a black hole has magnetic field F = (2a* + e" sin(2), y³ + xe*, 2z³ + e" cos(z) +…

A: TO FIND: Magnetic flux across the black hole

Q: Express the circular function cos+sin as a single cosine with a phase displacement T Rcos (+a) where…

A:

Q: If F(x, y) = (6xy - 1,3x2 + 2y) and C is the circular arc along x² + y? = 2 from (1, 1) to (-1, 1),…

A: Given vector field is  F(x,y)=6xy-1,3x2+2yF(x,y)=(6xy-1)i+(3x2+2y)j c is the circular arc along…

Q: Given: vector-valued function R(t) whose derivative R'(t) = (4t, 6t, —12t) (c) Determine the…

A:

Q: Find the circulation of the Fİeld F(x; y) = 3xi+yj around the circle r(t) = (cos6t)i + ( sin6t)j;…

A: We need to find the circulation of field F→=3xi^+yj^  around the circle r→t=cos6ti^+sin6tj^ where…

Q: Dircetional desivalive :- unut veetor giren flay)= e* siny by' ongle at (0,0) i) Q = ズ find Du f(do,…

A:

Q: 1 Consider the curve defined by the parametric equations x = ÷ť & y=÷ť for 2<t<3 2 3 Determine the…

A: take derivative

Q: 3. Find the arc length traced out by the endpoint of the vector-valued function 1 †(t) = t cos t î +…

A:

Q: If the electric intensity in the vacuum E = Eocosθar - Eosinθaθ is the divergence of the electric…

A: In the question it is asked to calculate the curl of electric intensity given.

Q: 13. Find the point on the curve 7(t) = (2 sin t, 2 cos t, t), 0<ts 7, where the tangent line is…

A:

Q: Suppose the divergence of the electric field in some spherical region is found to be V E = 5kr³.…

A: as given,   the divergence of electric field in some spherical region is ∇·E=5kr3 claim- to find the…

Q: A uniformly stretched string and fastened to two points 10 meters apart. The displacement of the…

A:

Q: Please solve

A: Parametrization of tangent lines is given by

Q: Find the motion of the mass-spring system in Fig. 72 with mass 0.125 kg, damping 0, spring constant…

A:

Q: Find the maximum, minimum and inflection points of the fur y= 3 sin 2x at [0, T].

A: Solution:-

Q: 1- f(-vt) ve x 1tve 117e x 1, wave which is mot a solilio to wave epuation. sinncost y at3 =…

A: We need to find the solution for equation y(x)=f(ax+bt)

Q: 3. Solve the Laplace equation inside a 60° wedge of radius 1 subjected to the boundary conditions…

A: The Laplace equation is given by: r2∂2u∂r2+r∂u∂r+∂2u∂θ2=0 The boundary conditions are: ∂u∂θr,0=0,…

Q: 3 2 Use a line integral to calculate the length of the helix ř(t) = (a cost, a sint, bt) 0<t < 2n…

A:

Q: Consider the parametric curve: X y = ln(-t) Which integral gives the arc length of the curve over…

A: We have to find the length of curve.

Q: Suppose F(2, y) = -y sin(æ)i + cos(2)j. (a) Find a vector parametric equation for the parabola y =…

A: Given F→(x,y) = -ysin(x)i→+cos(x)j→ a) Let us assume x=t then y=x2=t2 by differentiating r→(t) with…

Q: A uniform and stationary magnetic field Ris perpendicular to the plane containing a circular…

A:

Q: Find the approximate length of the polar curve r = e° from 0 = n to 0= 47; use any numerical…

A: The given polar curve is -  r = eθwhere θ varies from θ = π to θ = 4π We have to approximate the…

Q: Use Green's Theorem to evaluate the line integral. e cos(2y) dx - 2e* sin(2y) dy C: x2 + y? = a?

A:

Q: The area of the rotational surface resulting from rotating the arc of the curve x=cos^3 t, y=sin^3 t…

A:

Q: a) Without doing any calculations, state whether the vector field: er sin2(xy) F = -V 1 + x*y4 is…

A: (a) Since ∇×∇=0 So, curlF=0 Hence, F is conservative field.

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.