Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Determine whether or not
F is a conservative vector field. If it is, find a function
f such that
F=grad f. (If the vector field is not conservative, enter DNE.)

F(x,y)=◻)=(ye^(x)+sin(y))i+(e^(x)+x cos(y))j

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

View step-by-step help

View step-by-step help

Find derivatives of sine and cosine functions

Full solution

Q. Determine whether or not

\mathbf{F}

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

f

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

\newline

F(x, y)=\square)=\left(y e^{x}+\sin (y)\right) \mathbf{i}+\left(e^{x}+x \cos (y)\right) \mathbf{j}

Identify Components of Vector Field: Step $1$ : Identify the components of the vector field $F$ . $F(x, y) = (y e^x + \sin(y))\mathbf{i} + (e^x + x \cos(y))\mathbf{j}$
Check for Conservativity: Step $2$ : Check if the vector field $F$ is conservative by comparing the partial derivatives. $\newline$ For $F$ to be conservative, $\frac{\partial P}{\partial y}$ must equal $\frac{\partial Q}{\partial x}$ , where $P = y e^x + \sin(y)$ and $Q = e^x + x \cos(y)$ . $\newline$ $\frac{\partial P}{\partial y} = e^x + \cos(y)$ $\newline$ $\frac{\partial Q}{\partial x} = e^x - x \sin(y)$
Compare Partial Derivatives: Step $3$ : Compare the results from Step $2$ . $\newline$ Since $\partial P/\partial y = e^x + \cos(y)$ and $\partial Q/\partial x = e^x - x \sin(y)$ , they are not equal because of the $x \sin(y)$ term in $\partial Q/\partial x$ .

More problems from Find derivatives of sine and cosine functions

Question

Find the derivative of

f(x)

\newline

f(x) = \cos(x)

\newline

f'(x) =

Posted 1 month ago

Question

Find the derivative of

f(x)

\newline

f(x) = \tan(x)

\newline

f'(x) =

Posted 2 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = e^x

\newline

f'\left(x\right) =

Posted 2 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = \ln(x)

\newline

f'(x) =

Posted 2 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = \tan^{-1}{x}

\newline

f'(x) =

Posted 2 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = x e^x

\newline

f'\left(x\right) =

Posted 2 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = \frac{e^x}{x}

\newline

f'(x) =

Posted 2 months ago

Question

Find the derivative of

f(x)

\newline

f(x) = e^{(x + 1)}

\newline

f'\left(x\right) =

Posted 1 month ago

Question

Find the derivative of

f(x)

\newline

f(x) = \sqrt{x+3}

\newline

f'(x) =

Posted 2 months ago

Question

Find the derivative of

g(x)

\newline

g(x) = \ln(2x)

\newline

g^{\prime}(x) =

Posted 3 months ago