Title: Calculating Buffalo Movement using Flux and Green’s Theorem**Problem Description:**A buffalo stampede is described by a velocity vector field $\mathbf{F} = \langle xy - y^3, x^2 + y \rangle $ km/h in the region $D$ defined by $a \leq x \leq b$, $a \leq y \leq b$ in units of kilometers. Assuming a density $\rho = 510$ buffalo per square kilometer, with $a = 2$ and $b = 3$, use the Flux Form of Green's Theorem to determine the net number of buffalo leaving or entering $D$ per hour (equal to $\rho$ times the flux of $\mathbf{F}$ across the boundary of $D$).**Diagram Explanation:**1. **Left Figure:** An image showing a herd of buffalo in motion, depicted to illustrate the concept of a buffalo stampede. 2. **Right Figure:** A grid depicting vectors that visually represent the velocity vector field $\mathbf{F}$. The vectors display the direction and magnitude of the field across the defined region $D$.**Instructions:**Calculate the net number of buffalo moving across the boundary by applying the appropriate calculus techniques. Provide your answer as a whole number.**Input:**- **Net Number Box:** A designated space for entering the computed result, expressed as buffalo per hour.**Reference:**This exercise utilizes principles found in "Rogawski: Calculus Early Transcendentals" by W.H. Freeman.

The value of F: F=xy-y2i^+x2+yj^ According to Green Theorem, ∫y=ab∫x=ab∂M∂x-∂L∂y=∮ Ldx+M dy Putting…

A buffalo stampede (see the left figure) is described by a velocity vector field F = (xy -y',x + y) km/h in the region D defined by a < x< b, a sys b in units of kilometers (see the right figure). Assuming a density is p= 510 buffalo per square kilometer, a = 2 and b = 3, use the Flux Form of Green's Theorem to determine the net number of buffalo leaving or entering D per hour (equal to p times the flux of F across the boundary of D). (Give your answer as a whole number.)

A buffalo stampede (see the left figure) is described by a velocity vector field F = (xy -y',x + y) km/h in the region D defined by a < x< b, a sys b in units of kilometers (see the right figure). Assuming a density is p= 510 buffalo per square kilometer, a = 2 and b = 3, use the Flux Form of Green's Theorem to determine the net number of buffalo leaving or entering D per hour (equal to p times the flux of F across the boundary of D). (Give your answer as a whole number.)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Title: Calculating Buffalo Movement using Flux and Green’s Theorem

**Problem Description:**

A buffalo stampede is described by a velocity vector field $\mathbf{F} = \langle xy - y^3, x^2 + y \rangle $ km/h in the region $D$ defined by $a \leq x \leq b$, $a \leq y \leq b$ in units of kilometers. Assuming a density $\rho = 510$ buffalo per square kilometer, with $a = 2$ and $b = 3$, use the Flux Form of Green's Theorem to determine the net number of buffalo leaving or entering $D$ per hour (equal to $\rho$ times the flux of $\mathbf{F}$ across the boundary of $D$).

**Diagram Explanation:**

1. **Left Figure:** An image showing a herd of buffalo in motion, depicted to illustrate the concept of a buffalo stampede.

2. **Right Figure:** A grid depicting vectors that visually represent the velocity vector field $\mathbf{F}$. The vectors display the direction and magnitude of the field across the defined region $D$.

**Instructions:**

Calculate the net number of buffalo moving across the boundary by applying the appropriate calculus techniques. Provide your answer as a whole number.

**Input:**

- **Net Number Box:** A designated space for entering the computed result, expressed as buffalo per hour.

**Reference:**
This exercise utilizes principles found in "Rogawski: Calculus Early Transcendentals" by W.H. Freeman.

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