4.4.11 · D5Multivariable Calculus
Question bank — Gradient perpendicular to level curves - surfaces
Quick symbol refresh so nothing below is unearned:
- is a level curve: the set of input points where takes the same value . Walk along it and never changes.
- is the gradient: a vector in the input plane pointing the way climbs fastest.
- (dot product) means the two vectors are perpendicular. This is the whole engine, via Directional Derivative and the Multivariable Chain Rule.
True or false — justify
Along a level curve the value of is constant, so the Directional Derivative in the tangent direction is zero.
True — that zero directional derivative is exactly the statement that is perpendicular to the tangent, i.e. to the level curve.
always points in the direction the level curve runs.
False — it points across the level curve (perpendicular), toward increasing ; along the curve does not change, so the "most-change" arrow cannot lie along it.
The tangent line to has direction .
False — is the normal; the tangent direction is any nonzero vector perpendicular to , e.g. .
If at a point on the level set, the perpendicularity claim still tells us the tangent direction.
False — the zero vector is perpendicular to everything, so it selects no tangent direction; the statement becomes vacuous and the point may be a singular/critical point of the level set.
is perpendicular to the graph surface .
False — lives in the 2D input plane and is normal to the level curves there; a normal to the graph is , obtained from .
Scaling to rotates the gradient, so perpendicularity to the level curve could break.
False — points the same way (just longer), and the level curves are unchanged as sets, so it stays perpendicular.
Replacing by keeps the gradient pointing along the same line at a point where .
True — chain rule gives , a positive-or-zero scalar times , so its direction is unchanged (and perpendicularity survives) wherever .
Two different functions can have the exact same level curve through a point yet different gradient directions there.
False for the direction — both gradients are normal to the shared curve, so they lie along the same line; only their lengths and possibly signs (inward/outward) can differ.
Spot the error
" at : since , the tangent slope is ."
Error — is the normal, not the tangent. The tangent direction is perpendicular to , e.g. , giving slope , matching .
"To get the tangent plane to , use ."
Error — that formula gives the tangent to a level set . For the graph , rewrite as and use normal ; see Tangent Plane to a Surface.
" means ."
Error — the dot product is zero because is perpendicular to (tangent to the level set), not because vanishes.
"Since the gradient is perpendicular to the level curve, it must be perpendicular to the -axis too."
Error — perpendicularity is to the level curve at that point, whose direction generally is not the -axis; only if the curve happens to run along the -axis there would that follow.
"At a saddle-shaped contour crossing (where a level set self-intersects), is perpendicular to the level set."
Error — at such a point , so there is no well-defined single tangent direction; the perpendicularity statement requires .
"For at , points inward toward the origin."
Error — points radially outward, the direction of increasing ; grows as you leave the origin.
Why questions
Why does the Multivariable Chain Rule appear in the proof at all?
We differentiate the composition ; the chain rule is the only tool that converts "how changes" into "", linking function-change to motion.
Why must we use an arbitrary curve on the level set, not one specific curve?
Perpendicularity to the whole level set means perpendicularity to every tangent direction; a single curve only rules out one direction, so we need it to hold for all curves through the point.
Why does "" prove perpendicularity rather than just "no change"?
A zero dot product between nonzero vectors is the definition of perpendicular; since is the tangent, being orthogonal to it is exactly "normal to the level set".
Why does the same argument extend from level curves to level surfaces in 3D?
The proof never used the dimension: for any curve inside we still get , and enough such curves span the tangent plane, so is normal to the surface.
Why do Lagrange Multipliers set ?
Each gradient is normal to its own level set; at a constrained optimum the two level sets are tangent, so their normals point along the same line, i.e. .
Why is Steepest Descent / Gradient Descent guaranteed to move across contours, never along them?
It steps in , which is perpendicular to the level set; a step along a contour would have and reduce by nothing, so descent avoids it.
Edge cases
At a local maximum of , where , is the gradient perpendicular to the level set?
The statement is vacuous — the "level set" there can be a single point or degenerate, and the zero vector has no direction, so perpendicularity carries no information.
For a linear function , the level curves are parallel straight lines; is still normal to them?
Yes — is constant everywhere and is perpendicular to every one of the parallel lines ; a straight level "curve" is the simplest case, not an exception.
If a level curve has a sharp corner (from an implicit equation like ), does perpendicularity hold at the corner?
No — at a corner the tangent direction is undefined and is not differentiable there, so (hence the claim) does not exist at that point, only on the smooth pieces.
Consider , whose level set is the pair of vertical lines ; is perpendicular to them?
Yes — is horizontal, perpendicular to the vertical lines; the level set being disconnected does not break the local perpendicularity.
On the level set of (a parabola), does the perpendicularity depend on the value ?
No — the same chain-rule argument works for every ; at each point is normal to the parabola regardless of the level chosen.
What happens to the perpendicularity claim on a level set where is constant on a whole 2D region (a "flat" plateau)?
There throughout, so there is no distinguished normal direction; the level "set" is an area, not a curve, and the statement no longer applies.
Recall One-line summary of every trap here
is a normal vector in the input plane, defined and perpendicular to the level set only where is differentiable and ; it is never the tangent slope, never a normal to the graph, and its length can be rescaled without breaking the right angle.
Connections
- Directional Derivative — zero along the level set is the seed of perpendicularity.
- Multivariable Chain Rule — the engine behind .
- Tangent Plane to a Surface — the correct use of as a normal.
- Lagrange Multipliers — gradients align because level sets touch.
- Steepest Descent / Gradient Descent — steps perpendicular to contours.