To solve the given problem, we need to compute the magnitude of the vector field $\mathbf{F}(x, y)$ defined as:

$\mathbf{F}(x, y) = \mathbf{i} + \mathbf{j}.$

Step 1: Understand $\mathbf{F}(x, y)$

The vector $\mathbf{F}(x, y)$ can be expressed in component form: $\mathbf{F}(x, y) = (1, 1),$ where $1$ is the coefficient for $\mathbf{i}$ and $1$ is the coefficient for $\mathbf{j}$.

Step 2: Compute $\|\mathbf{F}(x, y)\|$

The magnitude of a vector $(a, b)$ is given by: $\|\mathbf{F}\| = \sqrt{a^2 + b^2}.$

Here, $a = 1$ and $b = 1$. Substituting these values: $\|\mathbf{F}\| = \sqrt{1^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2}.$

Final Answer:

$\|\mathbf{F}\| = \sqrt{2}.$

Would you like further clarification on this? Here are related questions to explore:

1. How is the magnitude of a vector defined mathematically?
2. What is the physical meaning of $\|\mathbf{F}(x, y)\|$ in this context?
3. How does the formula for the magnitude change if the vector field had components like $x$ or $y$?
4. What are the applications of vector magnitudes in physics and engineering?
5. Can this vector field $\mathbf{F}(x, y)$ vary depending on $x$ and $y$, and how would that affect the magnitude?

Tip: Always break down vector components clearly to avoid confusion during computations.