# Can there exist a $C^2$ function $f: \mathbb R^3 \to \mathbb R$ with \$f_x = 3x - 2y + 4z.

Can there exist a $$C^2$$ function $$f: \mathbb R^3 \to \mathbb R$$ with $$f_x = 3x - 2y + 4z, f_y = -2x+3y +z$$ and $$f_z = 3x + y - 5z$$. Explain your answer.

$$\int f_udu = \frac{3}{2}x^2 - 2xy + 4zu + f(y, z) = f(x, y, z)$$

$$\int f_y dy = -2xy + zy + \frac{3}{2}y^2 + f(x, z) = f(x, y, z)$$

$$\int f_z dz = 3xz + yz - \frac{5z^2}{2} + f(x, y) = f(x, y, z)$$

since $$\frac{d}{dz} (\int f_u du) \neq f_z$$ hence does not exist any for which the partial derivatives are given

is this correct?

There's a few different approaches. Yours is fine.

Hans Engler has a really good one. Here's a third, somewhere in between.

You can make a vector $$\vec{v}=$$

Then if $$\nabla \times \vec{v}=0$$ then such a function exists.

$$\nabla \times \vec{v}\implies v_i=\sum_{j=1}^3\sum_{k=1}^3 \epsilon_{ijk}\frac{\partial}{\partial x^j}\frac{\partial f}{\partial x_k}$$

Where $$\epsilon_{ijk}$$ is the Levi-Civita symbol. $$x_1=x, x_2=y, x_3=z.$$

You recover Mr. Engler's with perhaps a bit more of work.

Through abuse of notation you can find $$\nabla \times \vec{v}$$, the curl of $$\vec{v}$$ with a matrix.