Math / Calculus

QuestionIf $\int_{C} \vec{F} \cdot d \vec{r}=\int_{C}\left(-x-5 y^{3}\right) d x+\left(x^{3} y\right) d y$ , what is $\vec{F}$ ? $\vec{F}=$

Studdy Solution

STEP 1

1. The line integral is given in terms of its components.
2. The vector field $\vec{F}$ is expressed in terms of its components $F_1$ and $F_2$ .
3. The line integral $\int_{C} \vec{F} \cdot d \vec{r}$ can be expressed as $\int_{C} F_1 \, dx + F_2 \, dy$ .

STEP 2

1. Identify the components of the vector field $\vec{F}$ from the given line integral.
2. Write the vector field $\vec{F}$ in terms of its components.

STEP 3

The given line integral is:
$\int_{C}\left(-x-5 y^{3}\right) d x+\left(x^{3} y\right) d y$
From this, we can identify the components of the vector field $\vec{F}$ :
- The component $F_1$ that multiplies $dx$ is $-x - 5y^3$ . - The component $F_2$ that multiplies $dy$ is $x^3 y$ .

STEP 4

Write the vector field $\vec{F}$ using the identified components:
$\vec{F} = \langle F_1, F_2 \rangle = \langle -x - 5y^3, x^3 y \rangle$
The vector field $\vec{F}$ is:
$\boxed{\langle -x - 5y^3, x^3 y \rangle}$

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms