- a²(x²-y?) dx dy Exercise: Finish ter 8.
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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![### Double Integral and Gaussian Functions
On this page, we explore the evaluation of a double integral often encountered in multivariable calculus, particularly related to Gaussian functions.
#### Integral Expression
The integral presented is:
\[
\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-a^2 (x^2 + y^2)} \, dx \, dy
\]
This represents a double integral over the entire plane (from \(-\infty\) to \(\infty\) for both \(x\) and \(y\)) of a two-dimensional Gaussian function. The function inside the integral exponentiates a negative quadratic form of \(x\) and \(y\), scaled by a constant \(a\).
#### Task
**Exercise: Finish using polar coordinates**
The task involves converting the double integral from Cartesian coordinates \((x, y)\) to polar coordinates. This is a common technique used to simplify the evaluation of integrals involving radially symmetric functions.
**Steps to proceed:**
1. Convert \(x\) and \(y\) to polar coordinates:
- \(x = r\cos\theta\)
- \(y = r\sin\theta\)
2. Rewrite the integral using the Jacobian determinant for the change of coordinates, \(dx \, dy = r \, dr \, d\theta\).
3. Transform the integral limits:
- \(r \in [0, \infty)\)
- \(\theta \in [0, 2\pi]\)
4. Evaluate the integral in the new coordinates.
This transformation leverages the symmetry of the Gaussian function and often simplifies the integration process.
By performing this exercise, you'll gain a deeper understanding of coordinate transformations in multivariable calculus and techniques for evaluating complex integrals.
For more details on the steps involved in coordinate transformations and examples, please refer to the next section of our learning module:
- [Coordinate Transformations](#)
- [Evaluating Gaussian Integrals](#)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fca708747-373f-4dbe-b127-10492ca0e68f%2F104f0a28-fc97-4248-b1e2-70bfc034ddfc%2F6fnqnd_processed.png&w=3840&q=75)
Transcribed Image Text:### Double Integral and Gaussian Functions
On this page, we explore the evaluation of a double integral often encountered in multivariable calculus, particularly related to Gaussian functions.
#### Integral Expression
The integral presented is:
\[
\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-a^2 (x^2 + y^2)} \, dx \, dy
\]
This represents a double integral over the entire plane (from \(-\infty\) to \(\infty\) for both \(x\) and \(y\)) of a two-dimensional Gaussian function. The function inside the integral exponentiates a negative quadratic form of \(x\) and \(y\), scaled by a constant \(a\).
#### Task
**Exercise: Finish using polar coordinates**
The task involves converting the double integral from Cartesian coordinates \((x, y)\) to polar coordinates. This is a common technique used to simplify the evaluation of integrals involving radially symmetric functions.
**Steps to proceed:**
1. Convert \(x\) and \(y\) to polar coordinates:
- \(x = r\cos\theta\)
- \(y = r\sin\theta\)
2. Rewrite the integral using the Jacobian determinant for the change of coordinates, \(dx \, dy = r \, dr \, d\theta\).
3. Transform the integral limits:
- \(r \in [0, \infty)\)
- \(\theta \in [0, 2\pi]\)
4. Evaluate the integral in the new coordinates.
This transformation leverages the symmetry of the Gaussian function and often simplifies the integration process.
By performing this exercise, you'll gain a deeper understanding of coordinate transformations in multivariable calculus and techniques for evaluating complex integrals.
For more details on the steps involved in coordinate transformations and examples, please refer to the next section of our learning module:
- [Coordinate Transformations](#)
- [Evaluating Gaussian Integrals](#)
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