To identify the vector field $\mathbf{F}$ from the line integral expression, compare the given integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ with its expanded form $\int_C \mathbf{F} \cdot d\mathbf{r} = \int_C F_1 \, dx + F_2 \, dy$, where $\mathbf{F} = (F_1, F_2)$.

The integral provided is: $\int_C \mathbf{F} \cdot d\mathbf{r} = \int_C (5x + 3y^3) \, dx + (x^2y) \, dy.$

Here, by comparing terms:

• The coefficient of $dx$ is $F_1 = 5x + 3y^3$,
• The coefficient of $dy$ is $F_2 = x^2y$.

Thus, the vector field $\mathbf{F}$ is: $\mathbf{F} = (5x + 3y^3) \mathbf{i} + (x^2y) \mathbf{j}.$

Final Answer:

$\mathbf{F} = (5x + 3y^3, x^2y).$

Would you like further clarification or examples related to this?

Here are some related questions to consider:

1. What does the line integral $\int_C \mathbf{F} \cdot d\mathbf{r}$ represent geometrically?
2. How would you compute $\int_C \mathbf{F} \cdot d\mathbf{r}$ for a specific path $C$?
3. Can $\mathbf{F}$ be conservative? If so, how would you verify this?
4. How does the fundamental theorem for line integrals relate to this vector field?
5. Could you express the vector field in polar coordinates for specific applications?

Tip: When comparing coefficients in vector calculus, carefully match the terms with their corresponding $dx, dy, dz$ components to construct the vector field systematically.