We measure that with the dot product .
A vector like u = ( u 1 , u 2 ) u u
The magic case we need:
Intuition Why this is THE tool for "perpendicular""Perpendicular" is a statement about an angle , and the dot product is the one gadget that turns an angle into simple arithmetic (a 1 b 1 + a 2 b 2 ∇ f 0
Worked example Right-angle check( 6 , 8 ) ⋅ ( − 4 , 3 ) = ( 6 ) ( − 4 ) + ( 8 ) ( 3 ) = − 24 + 24 = 0. ( 6 , 8 ) ( − 4 , 3 )
Recall Edge case: the zero vectorIf ∇ f = ( 0 , 0 ) 0 everything , so "perpendicular" loses meaning. These are exactly the critical points , handled separately in Lagrange Multipliers and optimization. Away from them, the argument is clean.
Definition Parametrized curver ( t ) r ( t ) = ( x ( t ) , y ( t )) moving point whose location depends on time t t r ( t )
Definition Velocity / tangent vectorr ′ ( t ) r ′ ( t ) = ( x ′ ( t ) , y ′ ( t )) velocity arrow — it points in the direction of motion and is tangent to the path.
Picture it as an ant walking a route on the map. At any instant the ant faces some direction; that facing arrow is r ′ ( t )
Intuition Why the topic needs a moving pointTo say "∇ f a direction along the level curve . The trick: send the ant walking inside the level curve. Then its velocity r ′ ( t ) is a tangent-to-the-level-curve direction — precisely the thing we'll dot against ∇ f
The final tool. The ant walks the path r ( t ) height f ( r ( t ))
Intuition Why the chain rule, and why it seals the proofWe know two facts and need to connect them: (a) how the height depends on position (that's ∇ f how position depends on time (that's r ′ Multivariable Chain Rule is exactly the bridge that multiplies these together. Now the punchline: if the ant walks inside a level curve , its height never changes, so the left side is 0 ∇ f ⋅ r ′ = 0 ∇ f ⊥ r ′
This same dot product is also the definition of the Directional Derivative D u f = ∇ f ⋅ u
Function f of x and y as a height landscape
Level curve f = c, no change in height
Partial derivatives f_x and f_y
Gradient grad f, the uphill arrow
Zero dot product means perpendicular
Parametrized curve r of t
Tangent vector r prime of t
Chain rule d dt f of r = grad f dot r prime
Gradient perpendicular to level curves
What does f : R 2 → R A machine taking a point ( x , y )
What is a level curve f ( x , y ) = c All floor-spots at the same fixed height c f
How do you compute f x Freeze y f x
What is ∇ f The vector ( f x , f y )
What does a ⋅ b = 0 The two (nonzero) arrows are perpendicular, because cos 9 0 ∘ = 0
Compute ( 6 , 8 ) ⋅ ( − 4 , 3 ) − 24 + 24 = 0
What is r ′ ( t ) r ( t ) The velocity/tangent arrow, pointing along the direction of motion.
State the multivariable chain rule for d t d f ( r ( t )) ∇ f ( r ( t )) ⋅ r ′ ( t )
Why is ∇ f = ( 0 , 0 ) It is a critical (flat) point; its dot product with everything is 0