What is the relationship between gradient and surfaces in tensor calculus?

[Mathematicsresear]

<Moderator's note: Moved from a homework forum.>

1. Homework Statement
Is the gradient perpendicular to all surfaces or just level surfaces?
For instance, if I I have a function f(x,y)=z where z is the dependent variable then that is a surface, wouldn't that be a level surface to a function of x,y,z so shouldn't the gradient also be perpendicular to the surface and level surface?

 

[Orodruin]

For instance, if I I have a function f(x,y)=z where z is the dependent variable then that is a surface, wouldn't that be a level surface to a function of x,y,z

Yes. In particular, it would be the level surface $g(x,y,z) = 0$, where $g(x,y,z) = f(x,y) - z$.

so shouldn't the gradient also be perpendicular to the surface and level surface?

The gradient of $g(x,y,z)$ would be perpendicular to the surface. It is unclear what you mean by

Is the gradient perpendicular to all surfaces or just level surfaces?

 

[Mathematicsresear]

Yes. In particular, it would be the level surface $g(x,y,z) = 0$, where $g(x,y,z) = f(x,y) - z$.The gradient of $g(x,y,z)$ would be perpendicular to the surface. It is unclear what you mean by

I mean, would it also be perpendicular to the function g(x,y,z)?

 

[Orodruin]

You cannot just say "gradient". You must specify the gradient of what function.

 

[Charles Link]

If you take $g(x,y,z)=$ constant , the gradient of $g(x,y,z)$ is perpendicular to the surface defined by $g(x,y,z)=$ constant. It works also for the case of the constant equal to zero, where $g(x,y,z)=f(x,y)-z$. $\\$ In particular, $\nabla g(x,y,z)$ evaluated at $(x_o, y_o, z_o)$ is perpendicular to the surface $g(x,y,z)=g(x_o,y_o, z_o)$ at $(x_o,y_o, z_o)$.

 

[Mathematicsresear]

You cannot just say "gradient". You must specify the gradient of what function.

Alright, I understand. The gradient if perpendicular to a surface, but it also is pointing in the direction of greatest increase, I'm not sure what the link between those to are. So is it both, pointing in the direction of greatest increase, and perpendicular to the surface?

 

[Charles Link]

The reason for greatest increase is because $dg=\nabla g \cdot d \vec{s}=|\nabla g| \cos{\theta} |d \vec{s}|$, where $d \vec{s}=dx \, \hat{i}=dy \, \hat{j} +dz \, \hat{k}$. (Use the definition of the gradient and work out the partial derivatives, etc. and compute $\nabla g \cdot d \vec{s}$). $\\$ The dot product picks up a $\cos{\theta}$ factor that is equal to 1 if $d \vec{s}$ is parallel to $\nabla g$. $\\$ We can also compute $\frac{dg}{ds}=|\nabla g| \cos{\theta}$.

 

[RPinPA]

If you have a function $z = f(x,y)$, then the gradient is a 2-vector in (x,y) space. It lies in the (x,y) plane. It is perpendicular to the contour lines of z, which are curves in (x,y) space. But it makes no sense to talk about that vector being perpendicular to the surface.

With 2 variables you can envision this very easily as a surface, for instance a hillside, on the surface of the earth. We'll pretend the Earth is flat. Imagine you are standing on a hillside, and your (x,y) directions are East and North. Look around the hill from where you are standing. One direction goes most steeply up the hill, assuming you're not at a local maximum. That's the direction of the gradient of your hill surface. It is a compass direction, like "northeast".

Is "northeast" perpendicular to the hillside? Is that a meaningful question?

 

[Charles Link]

@RPinPA makes a good point that was previously omitted/overlooked: $\\$ $dz=(\frac{\partial{f}}{\partial{x}}) \, dx+(\frac{\partial{f}}{\partial{y}}) \, dy =\nabla^{(2)} f \cdot d \vec{r}=|\nabla^{(2)} f| \cos(\theta) | d \vec{r}|$, where $d \vec{r} =dx \, \hat{i}+ dy \, \hat{j}$. $\\$ Here $\nabla^{(2)}$ refers to a two-dimensional gradient. $\\$ The previous 3 dimensional gradient is perpendicular to the surface of the "hill" $f(x,y)-z=0$. Meanwhile, my post 7 is correct in regards to the function $g$, but doesn't correctly answer the question in regards to height $z$.

 

[wrobel]

Offtop for those who started to study tensor calculus

It is important to add that one should not be confused with two different objects
1) differential of a function which is a covector with components $(\frac{\partial f}{\partial x^i})$
and
2) gradient of a function $(\nabla f)^i=g^{ij} \frac{\partial f}{\partial x^j}$ which is a vector
These are two different types of tensors but their components coincide in Cartesian frame as long as it exists. Here $g^{ij}$ is the inverse Gramian matrix