Using the transformation x + y = u, y = v. The value of Jacobian (J) for the integralisa)zerob)1c)2d)4Correct answer is option 'B'. Can you explain this answer?

EduRev

All Courses

Explore Courses Mathemat... Exam Mathemat... Test Series Free Notes EduRev Infinity

Open App

Mathematics Exam > Mathematics Questions > Using the transformation x + y = u, y = v. Th...

Using the transformation x + y = u, y = v. The value of Jacobian (J) for the integral

a)
zero
b)
1
c)
2
d)
4

Correct answer is option 'B'. Can you explain this answer?

Join for Free

Join for Free

Verified Answer

Using the transformation x + y = u, y = v. The value of Jacobian (J) f...

To solve this problem, we need to evaluate the integral:

∫₀¹ ∫₀¹ |cos(x + y)| dx dy

using the transformation x + y = u and y = v.

Step 1: Determine the Transformation and Jacobian

The transformation is given by:

u = x + y
v = y

To express x and y in terms of u and v:

From u = x + y, we can solve for x as x = u - v.
Thus, x = u - v and y = v.

Now, we need to calculate the Jacobian J of the transformation:

Calculating each partial derivative:

∂x/∂u = 1
∂x/∂v = -1
∂y/∂u = 0
∂y/∂v = 1

So, the Jacobian is:

J = | 1 -1 | = (1)(1) - (-1)(0) = 1.
| 0 1 |

Thus, the Jacobian J = 1.

Step 2: Set Up the Integral in Terms of u and v

Since J = 1, the integral becomes:

∫₀¹ ∫₀¹ |cos(u)| du dv.

Step 3: Evaluate the Integral

We can separate the integrals because |cos(u)| depends only on u:

∫₀¹ ∫₀¹ |cos(u)| du dv = ∫₀¹ ( ∫₀¹ |cos(u)| du ) dv.

First, we evaluate the inner integral with respect to u:

∫₀¹ |cos(u)| du.

Since cos(u) is positive over the interval [0, 1], we have:

∫₀¹ cos(u) du = [sin(u)]₀¹ = sin(1) - sin(0) = sin(1).

Now, substitute this result back:

∫₀¹ ∫₀¹ |cos(u)| du dv = ∫₀¹ sin(1) dv = sin(1) * [v]₀¹ = sin(1).

Conclusion:

The value of the integral is approximately: B: 1.

View all questions of this test

Most Upvoted Answer

Using the transformation x + y = u, y = v. The value of Jacobian (J) f...

Answer this doubt

Explore Courses for Mathematics exam

Similar Mathematics Doubts

Using the transformationx + y = u, y = v. The value of Jacobian (J) for the integralis

View answer

Consider the linear transformationfiven byL (x ,y ,z ) = (k x + y + z , x + ky + z , x + y + kz), If P be the matrix representation of L relative to the standard basis. Then the value of K for which P has zero as one of the eigen value is ________. ( where k < 0) Correct answer is '-2'. Can you explain this answer?

View answer

The value of double integralsover region common to the circlesx2 + y2 = x, x2 + y2 = y i s ______ . (correct upto four decim al p la c e s ) Correct answer is '0.0104'. Can you explain this answer?

View answer

The transformation T:R^(3)rarr R^(2) defined T(x y z)=(x y y z) is (

View answer

What is Eigen value? Correct answer is 'A scalar associated with a given linear transformation'. Can you explain this answer?

View answer

Question Description
Using the transformation x + y = u, y = v. The value of Jacobian (J) for the integralisa)zerob)1c)2d)4Correct answer is option 'B'. Can you explain this answer? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Using the transformation x + y = u, y = v. The value of Jacobian (J) for the integralisa)zerob)1c)2d)4Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Using the transformation x + y = u, y = v. The value of Jacobian (J) for the integralisa)zerob)1c)2d)4Correct answer is option 'B'. Can you explain this answer?.

Solutions for Using the transformation x + y = u, y = v. The value of Jacobian (J) for the integralisa)zerob)1c)2d)4Correct answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mathematics. Download more important topics, notes, lectures and mock test series for Mathematics Exam by signing up for free.

Here you can find the meaning of Using the transformation x + y = u, y = v. The value of Jacobian (J) for the integralisa)zerob)1c)2d)4Correct answer is option 'B'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of Using the transformation x + y = u, y = v. The value of Jacobian (J) for the integralisa)zerob)1c)2d)4Correct answer is option 'B'. Can you explain this answer?, a detailed solution for Using the transformation x + y = u, y = v. The value of Jacobian (J) for the integralisa)zerob)1c)2d)4Correct answer is option 'B'. Can you explain this answer? has been provided alongside types of Using the transformation x + y = u, y = v. The value of Jacobian (J) for the integralisa)zerob)1c)2d)4Correct answer is option 'B'. Can you explain this answer? theory, EduRev gives you an ample number of questions to practice Using the transformation x + y = u, y = v. The value of Jacobian (J) for the integralisa)zerob)1c)2d)4Correct answer is option 'B'. Can you explain this answer? tests, examples and also practice Mathematics tests.

Explore Courses for Mathematics exam