Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Determine whether or not $\newline$ $F$ is a conservative vector field. If it is, find a function $\newline$ $f$ such that $\newline$ F=\(\newlineabla f\). (If the vector field is not conservative, enter DNE.) $\newline$ $f(x,y)=\square(x,y)=(ye^{x}+\sin(y))i+(e^{x}+x \cos(y))j$

View step-by-step help

View step-by-step help

Write equations of cosine functions using properties

Full solution

Q. Determine whether or not

\newline

F

is a conservative vector field. If it is, find a function

\newline

f

such that

\newline

F=\(\newlineabla f\). (If the vector field is not conservative, enter DNE.)

\newline

f(x,y)=\square(x,y)=(ye^{x}+\sin(y))i+(e^{x}+x \cos(y))j

Check for Conservative: To check if $F$ is conservative, we need to verify if the curl of $F$ is zero. The vector field $F$ is given by $F = (y e^x + \sin(y))\mathbf{i} + (e^x + x \cos(y))\mathbf{j}$ . Calculate the partial derivatives: $\partial/\partial y$ of $(y e^x + \sin(y)) = e^x + \cos(y)$ , $\partial/\partial x$ of $(e^x + x \cos(y)) = e^x - x \sin(y)$ .
Calculate Partial Derivatives: Compare the partial derivatives: $\frac{\partial}{\partial y}$ of the $i$ -component = $e^x + \cos(y)$ , $\frac{\partial}{\partial x}$ of the $j$ -component = $e^x - x \sin(y)$ . Since these are not equal, the curl of $F$ is not zero.
Compare Partial Derivatives: Since the curl of $\mathbf{F}$ is not zero, $\mathbf{F}$ is not a conservative vector field.

More problems from Write equations of cosine functions using properties

Question

-9 i \cdot(6 i+8)=

\newline

Your answer should be a complex number in the form

a+b i

a

b

are real numbers.

Posted 3 months ago

Question

15 \cdot(-4 i+2)=

\newline

Your answer should be a complex number in the form

a+b i

a

b

are real numbers.

Posted 3 months ago

Question

-4 \cdot(20 i+10)=

\newline

Your answer should be a complex number in the form

a+b i

a

b

are real numbers.

Posted 3 months ago

Question

13 i \cdot(1-5 i)=

\newline

Your answer should be a complex number in the form

a+b i

a

b

are real numbers.

Posted 3 months ago

Question

14 \cdot(-2 i+6)=

\newline

Your answer should be a complex number in the form

a+b i

a

b

are real numbers.

Posted 3 months ago

Question

10 \cdot(-6-9 i)=

\newline

Your answer should be a complex number in the form

a+b i

a

b

are real numbers.

Posted 3 months ago

Question

-8 i \cdot(12-i)=

\newline

Your answer should be a complex number in the form

a+b i

a

b

are real numbers.

Posted 3 months ago

Question

-20 \cdot(2 i-7)=

\newline

Your answer should be a complex number in the form

a+b i

a

b

are real numbers.

Posted 3 months ago

Question

-3 i \cdot(8 i+5)=

\newline

Your answer should be a complex number in the form

a+b i

a

b

are real numbers.

Posted 3 months ago

Question

-12 \cdot(8-2 i)=

\newline

Your answer should be a complex number in the form

a+b i

a

b

are real numbers.

Posted 3 months ago