Revision history [back]

So I feel like I should know this but...

In three dimensional space the del operator is defined as $\nabla =\langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \rangle$. So when dealing with a function like the one in problem 4 of the review sheet; $f(x,y)=xy^{3}$. Do we treat del as being $\nabla=\langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y} \rangle$ and find a gradient with two components or do remake the function into a level surface such as $xy^{3}-z=k $ where $k \in \mathbb{Z} $ and find a gradient with three components?

If this is obvious I'm incredibly sorry but it really is bugging me.

So I feel like I should know this but...

In three dimensional space the del operator is defined as $\nabla =\langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \rangle$. So when dealing with a function like the one in problem 4 of the review sheet; $f(x,y)=xy^{3}$. Do we treat del as being $\nabla=\langle \frac{\partial}{\partial x}, \frac{\partial}{\partial y} \rangle$ and find a gradient with two components or do we remake the function into a level surface such as $xy^{3}-z=k $ where $k \in \mathbb{Z} $ and find a gradient with three components?

If this is obvious I'm incredibly sorry but it really is bugging me.