Evaluating the gradient at every point fills the plane with arrows. The result is a gradient field, and it always flows uphill.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The gradient is a normal
Since ∇f is perpendicular to the level curve, it serves as a normal vector for that curve at the point.
Tangent lines the easy way
With normal ⟨A, B⟩ at (a, b), the tangent line to the level curve is A(x − a) + B(y − b) = 0. No solving for y is needed.
In three dimensions
For a surface written as F(x, y, z) = k, the gradient ∇F is normal to the surface. The tangent plane follows immediately.
A sphere
For x² + y² + z² = R² the gradient is ⟨2x, 2y, 2z⟩, which points straight out from the centre. That is exactly the radius direction.
A preview
Not every field of arrows is a gradient field. Which ones are becomes the central question of Chapter 8.
The gradient drawn everywhere at once
Computing the gradient at every point produces a vector field of arrows. Their directions show where the function increases fastest and their lengths show how fast.
Normals to surfaces
For a surface defined by F(x, y, z) = 0, the gradient of F is normal to it. That is the standard route to a tangent plane for a surface not given as z = f(x, y).
Gradient fields are special
Not every vector field is the gradient of something. Those that are behave far better in line integrals, which is what the conservative field lesson later returns to.
Forces as gradients
Gravitational and electrostatic forces are gradients of potential functions, with a minus sign. That is why potential energy is such a useful concept — one scalar encodes a whole force field.
Step 2: Try It Yourself
Tap and try it out.
- Released from(1, 1)
Every arrow points away from the origin, so trajectories run outward. The origin is unstable.
Step 3: Watch an Example
One step at a time.
Watch Grace Find a Tangent Line
Grace wants the tangent line to x² + y² = 25 at the point (3, 4).
- Step 1
She treats the circle as a level curve of f = x² + y².
Step 4: Your Turn
Practice makes it stick.
The Sphere
Problem 1 of 2
On x² + y² + z² = 14 at (1, 2, 3), the gradient is ⟨2x, 2y, 2z⟩. What is its z-component there?
The Normal
Problem 2 of 2
A level curve has normal ⟨4, 3⟩ at a point. What is the magnitude of that normal?
Normals Everywhere
1 of 8
f = x² + y². What is the x-component of ∇f at (5, 2)?
2 of 8
Is the gradient perpendicular to the level curve? 1 yes, 0 no.
3 of 8
f = 3x + 4y. What is the magnitude of ∇f?
4 of 8
The tangent line 6(x − 3) + 8(y − 4) = 0 expands to 6x + 8y = k. What is k?
5 of 8
On a sphere, does the gradient point along the radius? 1 yes, 0 no.
6 of 8
f = xy. What is the y-component of ∇f at (4, 9)?
7 of 8
Order the steps for finding a tangent line to a level curve.
- 1Compute the gradient of f
- 2Evaluate the gradient at the point
- 3Use it as the normal in the line equation
- 4Write the curve as a level curve of some f
8 of 8
f = 7. What is the magnitude of ∇f?
Step 5: Quick Check
Show what you know.
Question 1 of 2
f = x² + y² at (6, 8). What is the magnitude of ∇f?
Question 2 of 2
Why does the gradient work as a normal vector to a level curve?
What You Learned
- The gradient evaluated everywhere gives a field of arrows all flowing uphill.
- Because it is perpendicular to level curves, the gradient is a ready-made normal vector.
- That makes tangent lines and tangent planes fall out with no rearranging.