Answer this Vector Calculus Problems ### Understanding Gradient Directions from Contour DiagramsA contour diagram of \( z = f(x, y) \) is provided to analyze the partial derivatives \( f_x \) and \( f_y \) and determine if they are positive, negative, or zero at specific points labeled P, Q, R, and S.#### Analysis Tasks:1. **Evaluate \( f_x \) at point P.**2. **Evaluate \( f_y \) at point P.**3. **Evaluate \( f_x \) at point Q.**4. **Evaluate \( f_y \) at point S.**5. **Evaluate \( f_y \) at point Q.**6. **Evaluate \( f_y \) at point R.**7. **Evaluate \( f_x \) at point S.**8. **Evaluate \( f_x \) at point R.**#### Explanation of the Contour Diagram- **Contour Levels**: The diagram consists of several contour lines representing levels of constant \( z \). These lines form concentric circles.- **Gradient and Direction**: - If moving to the right on the x-axis increases the contour level (darker regions), \( f_x \) is positive. - If the contour level decreases, \( f_x \) is negative. - Similarly, if moving up the y-axis increases the level, \( f_y \) is positive, and if it decreases, \( f_y \) is negative.- **Points of Interest**: - **P, Q, R, S**: Points marked on the plot where you will evaluate the signs of partial derivatives. - **Color Indication**: - **"Low" to "High"**: The color gradient from dark to light indicates the increasing value of \( z \).This contour analysis aids in understanding how changes in \( x \) or \( y \) affect \( z \) at various points in the diagram, providing insights into the behavior of functions represented in a two-dimensional plane.

1. Negative 2. Positive 3. Negative 4. Positive 5. Negative 6. Positive 7. Positive…

A contour diagram of z = f(x, y) is given in the figure. Determine whether fx and fy are positive, negative, or zero at the points P, Q, R, and S. ? ? ? ? ► 1. fx at the point P. 2. fy at the point P. 3. fx at the point Q. 4. f., at the point S. high high рәш S. high high (Click on graph to enlarge)

A contour diagram of z = f(x, y) is given in the figure. Determine whether fx and fy are positive, negative, or zero at the points P, Q, R, and S. ? ? ? ? ► 1. fx at the point P. 2. fy at the point P. 3. fx at the point Q. 4. f., at the point S. high high рәш S. high high (Click on graph to enlarge)

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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Answer this Vector Calculus Problems

### Understanding Gradient Directions from Contour Diagrams

A contour diagram of \( z = f(x, y) \) is provided to analyze the partial derivatives \( f_x \) and \( f_y \) and determine if they are positive, negative, or zero at specific points labeled P, Q, R, and S.

#### Analysis Tasks:
1. **Evaluate \( f_x \) at point P.**
2. **Evaluate \( f_y \) at point P.**
3. **Evaluate \( f_x \) at point Q.**
4. **Evaluate \( f_y \) at point S.**
5. **Evaluate \( f_y \) at point Q.**
6. **Evaluate \( f_y \) at point R.**
7. **Evaluate \( f_x \) at point S.**
8. **Evaluate \( f_x \) at point R.**

#### Explanation of the Contour Diagram
- **Contour Levels**: The diagram consists of several contour lines representing levels of constant \( z \). These lines form concentric circles.
- **Gradient and Direction**:
- If moving to the right on the x-axis increases the contour level (darker regions), \( f_x \) is positive.
- If the contour level decreases, \( f_x \) is negative.
- Similarly, if moving up the y-axis increases the level, \( f_y \) is positive, and if it decreases, \( f_y \) is negative.

- **Points of Interest**:
- **P, Q, R, S**: Points marked on the plot where you will evaluate the signs of partial derivatives.

- **Color Indication**:
- **"Low" to "High"**: The color gradient from dark to light indicates the increasing value of \( z \).

This contour analysis aids in understanding how changes in \( x \) or \( y \) affect \( z \) at various points in the diagram, providing insights into the behavior of functions represented in a two-dimensional plane.

Calculus that deals with differentiation and integration of the vector field in three-dimensional Euclidean space. It deals with quantities that have both magnitude and direction.

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