Suppose we have a scalar field $T$ and we take a point in space, say $(\alpha,\beta,\gamma)$. The direction of the gradient $\nabla T$ at $(\alpha,\beta,\gamma)$ gives us the direction along which the maximum change in $T$ will exist, infinitesimally. We take some infinitesimal length $\mathrm dl$ and consider $\theta$ to be the angle between $\nabla T$ and $\vec{\mathrm dl}$. Now for a fixed value of $\theta$, the set of points that we get will form a cone with $(\alpha,\beta,\gamma)$ as it's apex and $\nabla T$ along it's axis.
The value of $\mathrm dT=(\nabla T)\cdot(\vec{\mathrm dl}) = |\nabla T||\mathrm dl|\cos\theta$ will remain the same for a fixed value of $\theta$
Does this mean that the value of $T$ will be the same at all points on the circumference of the cone's base? For all scalar fields $T$?
Is this true or did I make some mistake? The result feels a little bizarre to me.