Q#02;Evaluate |G(x, y)dx, [ G(x,y)dy and [ G(x, y)ds on the indicated curve C. G (x, y) = x'y² + y² sin 2x + cos 2x, x = 2t, y = t,0

Q: Q1// B: If u+v=e cos y and u-v=e'sin yfind the Jacobain of the a u, v at (x.y) (1,0) dx,y function

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Q: *Gl0/ Apply Stoke's theorem to Cal cu late S4y dx +2zdy + 6y dz where c i$ the curve of xyt 2 = 6z…

A: Apply Stokes theorem to calculate ∫C(4ydx+2xdy+6ydz) where c: intersection of x2+y2+z2=6z and z=x+3…

Q: 2: Evaluate [G(x, y)dx, [G(x,y)dy and [G(x, y)ds on the indicated curve C. G(x, y) = x³y² + y² sin…

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Q: a) Evaluate Vx² + y² dydx. 1 J-Vi b) Evaluate the line integral | cos x cos y da + (1 – sin æ sin y)…

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Q: When is the derivative dy/dx of the parametric equations x(t)= 3cos t and y(t)=3sin t for Ost22n…

A: Let's find.

Q: Example: » - ( ) F(r,y) = | x² + y² ' x² + y², on the curve C : R(t) = (cos t, sin t), t e [0, 27].…

A: Given : vector field F→x, y=-yx2+y2, xx2+y2 on the curve C : R→t=cos t, sin t , t∈0, 2π

Q: Use the differential dy to approximate Ay when x changes as indicated. y = V3x – 2; from x = 2 to x…

A: Topic - approximation Using derivatives

Q: 3.) Calculate the curve integral f (1 – a² + y²) dx + (1 + x² – y°) dy. Here C consists of the…

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Q: Use Green's theorem to evaluate S. (5xy + x² + y²) dx + (x² – y)dy where C is a closed curve that is…

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Q: S VF-1 The arc length of the curve y(x) ² – 1 dt for x E [1,9] is

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Q: Use the differential dy to approximate 4ywhen x changes as indicated. y = V3x – 2; from x = 2 to x =…

A: y= 3x-2Find dy/dx when x changes from x=2 to x= 2.03

Q: F(x,y) = (4xy^3 + 9)i + (6x^2*y^2+2*e^(2y))j  conservative by finding a potential function f for F,…

A: let f be the potential function

Q: (xcos"(yz- my XZ -1 x cos If w= In i+ mxy' j+ y(z-mx)k and f =w find at point (-m.2.0): (1) the…

A: Answer

Q: HW 0 1f y=tanx show thats y" = -2Siny Cosy %3D

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Q: (0. ) : If f(x, y) = 12e cos x + 3 cot y, then the total differential of f at is : J, df = -12 dx 6…

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Q: Which of the following expressions are true? 1. 2z + 4y – z = 5 is the equation of a tangent plane…

A: The equation of a tangent to a function is given by the formula . Here,  is the point at which the…

Q: Q4. Let C be a portion of a curve from (2,0) to (2, 2). Let F(r, y) = (tan(2²) + e²", 2.re²y + cos…

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Q: (x,() + x(x, t) = sin(t) u(n, 0) = cos(7), at %3D %3D nER.

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Q: Use Green's theorem to evaluate (2y dx-x dy) around the closed curve C. 27 (2,8) 8. 4. C. -2 (1,-1)

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Q: Q2// B :- If u+v3e"cos y and u- v=e' sin y find the Jacobain of the function alu,v at dx.yl (x.y)…

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Q: 2. (i) Find dy where dx (а) In(ху) %3D х- у 1-x (b) 1+x dy (ii) Find the maximum value of the curve…

A: Given that  The following equations  To find the derivative

Q: (a) Find the differential dy. y = 5 tan x dy = (b) Evaluate dy for the given values of x and dx. x =…

A: Formula used:  ddxa.x=a.ddxx ddxtan x= sec2x

Q: .: Evaluate [G(x, y)dx, [G(x,y)dy and [G(x, y)ds on the indicated curve C. G (x, y) = x³ y² + y² sin…

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Q: -2y The line integral (e sin x) dx+(x' +3y)dy with path F (t)= (8cost, 8 sint) such that 0sts2n. Set…

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Q: Q.3 a) Verify Green's theorem for |(x2 + 2xy) dx + (y? + x³y) dy Where c is a square with vertices…

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Q: Which of the following expressions are true? 1. 2x + 4y – z = 5 is the equation of a tangent plane…

A: Hence option (c) is correct.

Q: 2. Find the work done by F in moving a particle once counterclockwise around the given curve C F =…

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Q: Integrate f(x,y) = x-5y over the curve C: x +y = 64 in the first quadrant from (0,8) to (V32,/32).…

A: Formula Used:Line integral ∫Cf(x, y) ds=∫abf(x(t), y(t)) dxdt2+dydt2dt

Q: 3.) Calculate the curve integral (1 – x² + y²) dx + (1 + 2² – y²) dy. Here C consists of the curves…

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Q: 3. Find the work done by F over the curve in the direction of increasing t. = (2,x, y) = (sin t, cos…

A: Write the vector field F=zi+xj+yk The equation of the curve. rt=sinti+costj+tk Differentiate…

Q: Consider the closed curve in the xy-plane given by x² + 2x + y* + 4y = 5. dy -(x+1) (a) Show that dx…

A: Given data:   The expression for the given function is x2+2x+y4+4y=5.   (a) Differentiate the above…

Q: Integrate f(x,y) =x -y over the curve C: x² + y = 16 in the first quadrant from (0,4) to (V8,18 - y)…

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Q: Exercise III. Find the length of the curve a) x = t³ y = t2 0<t< 4 b) x = cost y = sint 0 <t< 2n c)…

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Q: Let y = tan(3r + 4). Find the differential dy when z = 2 and dr 0.1 %3D Find the differential dy…

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Q: 2. Let F(x,y) = (cos(x – 2y), –2 cos(x – 2y)). Find curves C and C2 so | that, F. dr = 0 F. dr = 1

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Q: (dy = arctan dx, (i) Use the facts: dy and (dy/dt) to find dt d dx (dx/dt) ds (ii) Use the fact that…

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Q: 1. If F(x, y) = (3+2xy)i + (x² – 3y²)j. Evaluate F. dr, where C is the curve given by r(t) = e'…

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Q: Evaluate the line integral: [ y dx + x dy ; C is the curve x 1 = t² - 1, y = 6t, 0 < t s 1

A: Note: - Since you have asked multiple questions, we will solve the first question for you. If you…

Q: (2² + y² – 1)³ – x²y3 = 0, 2- Given the curve dy find dx -2 2

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Q: [&+2y)dx +(x² - 3y°)dy (3+2 4. Evaluate where C is any smooth curve from the point A (1, 0) to the…

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Q: (a) Find the differential dy. y = V 24 + x dy = (b) Evaluate dy for the given values of x and dx.…

A: Topic = Derivative

Q: (a) Find the differential dy. y = V3 + x2 dy = (b) Evaluate dy for the given values of x and dx. x =…

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Q: 2. If F(x, y, z) = y²i+(2xy+e3=)j+3ye³=k. Evaluate F·Î ds, where C is the curve represented by r(t)…

A: Given F→x,y,z=y2i^+2xy+e3zj^+3ye3zk^ and the curve is C given by r^t=cost,t,sint,  0≤t≤π2. Find…

Q: Show that the curve r(t) = ( t3 /4 - 2)i +(4/t-3)j+cos(t-2)k is tangent to the surface x3 + y3 + z3…

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Q: (a) Find the differential dy. y = V 24 + x2 dy = (b) Evaluate dy for the given values of x and dx. X…

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Q: Use Green's theorem to evaluate S. (5xy + x² + y²) dx + (x² – y)dy where C is a closed curve that is…

A: According to the given information, it is required to use green’s theorem to evaluate the integral.

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.