REFER TO IMAGE ---### Understanding the Divergence of a Vector FieldIn this section, we will explore the concept of divergence in the context of a given vector field. Let’s analyze the vector field $ \mathbf{F}(x, y) = x^2 \hat{\mathbf{i}} + y^2 \hat{\mathbf{j}} $.#### Vector Field VisualizationBelow is a graph representing this vector field:![Vector Field Graph](vector_field_graph.png)This graph displays arrows indicating the direction and magnitude of the vector field at different points in the plane. The arrows point outward from the origin, with their lengths increasing as they move away from the origin, particularly emphasizing the behavior of $ \mathbf{F}(x, y) $.#### Problems and Solutions(a) **Calculate $\operatorname{div} \mathbf{F}$:** To find the divergence of the vector field $ \mathbf{F}(x, y) = x^2 \hat{\mathbf{i}} + y^2 \hat{\mathbf{j}} $: \[ \operatorname{div} \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} \] Here, $ F_1 = x^2 $ and $ F_2 = y^2 $. \[ \operatorname{div} \mathbf{F} = \frac{\partial x^2}{\partial x} + \frac{\partial y^2}{\partial y} = 2x + 2y \](b) **For which points $(x, y)$ do we have $\operatorname{div} \mathbf{F} = 0$? How can we visualize this?** Setting the divergence to zero, we get: \[ 2x + 2y = 0 \implies x + y = 0 \] These points lie along the line $ y = -x $.(c) **For which points $(x, y)$ do we have $\operatorname{div} \mathbf{F} > 0$? How can we visualize this?** For the divergence to be positive: \[ 2x + 2y > 0 \implies x +

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Description: REFER TO IMAGE ---### Understanding the Divergence of a Vector FieldIn this section, we will explore the concept of divergence in the context of a given vector field. Let’s analyze the vector field \( \mathbf{F}(x, y) = x^2 \hat{\mathbf{i}} + y^2 \ha

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- Here is a graph of the vector field F(r, y) = r²i+ y°j. 2 (a) Calculate div F. (b) For which points (r, y) do we have div F = 0? How can we visualize this? %3D

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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---

### Understanding the Divergence of a Vector Field

In this section, we will explore the concept of divergence in the context of a given vector field. Let’s analyze the vector field $ \mathbf{F}(x, y) = x^2 \hat{\mathbf{i}} + y^2 \hat{\mathbf{j}} $.

#### Vector Field Visualization

Below is a graph representing this vector field:

![Vector Field Graph](vector_field_graph.png)

This graph displays arrows indicating the direction and magnitude of the vector field at different points in the plane. The arrows point outward from the origin, with their lengths increasing as they move away from the origin, particularly emphasizing the behavior of $ \mathbf{F}(x, y) $.

#### Problems and Solutions

(a) **Calculate $\operatorname{div} \mathbf{F}$:**

To find the divergence of the vector field $ \mathbf{F}(x, y) = x^2 \hat{\mathbf{i}} + y^2 \hat{\mathbf{j}} $:

\[
\operatorname{div} \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y}
\]

Here, $ F_1 = x^2 $ and $ F_2 = y^2 $.

\[
\operatorname{div} \mathbf{F} = \frac{\partial x^2}{\partial x} + \frac{\partial y^2}{\partial y} = 2x + 2y
\]

(b) **For which points $(x, y)$ do we have $\operatorname{div} \mathbf{F} = 0$? How can we visualize this?**

Setting the divergence to zero, we get:

\[
2x + 2y = 0 \implies x + y = 0
\]

These points lie along the line $ y = -x $.

(c) **For which points $(x, y)$ do we have $\operatorname{div} \mathbf{F} > 0$? How can we visualize this?**

For the divergence to be positive:

\[
2x + 2y > 0 \implies x +

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