Difficulty: Intermediate | Prerequisites: partial differentiation basics, vectors and planes in 3D
This material ties together the geometric and analytic sides of functions of two or three variables. Level sets show you what a function looks like without plotting it in higher dimensions. Tangent planes are the multivariable version of tangent lines, and classifying quadratic surfaces rounds out the visual toolkit for this part of the course.
Level sets of a function T(x, y, z) are the surfaces (or curves, in 2D) where T takes a constant value. For a linear function these are parallel planes. The tangent plane to z = f(x, y) at a point is the best flat approximation to the surface there, built from the partial derivatives. Quadratic surfaces (paraboloids, saddles) are classified by looking at the signs and cross-terms in the quadratic form.
Level set
The set of all points where a function equals a constant: {(x, y, z) : T(x, y, z) = c}. For a function of three variables, each level set is a surface. For a function of two variables, each level set is a curve.
In simple terms, level sets are the contour lines (or contour surfaces) of a function.
Partial derivative
f_x(a, b) is the rate of change of f with respect to x at the point (a, b), holding y fixed. Similarly for f_y.
Think of it as: slice the surface with a plane parallel to the xz-plane (fixing y), then take the ordinary derivative of the resulting curve.
Tangent plane
The plane that best approximates the surface z = f(x, y) near a point (a, b, f(a, b)). Its equation is z − f(a, b) = f_x(a, b)(x − a) + f_y(a, b)(y − b).
In simple terms, this is the flat sheet that just touches the surface at one point and has the same slope in every direction there.
Section (cross-section) of a surface
The curve you get by intersecting a surface z = f(x, y) with a plane such as x = constant. On the plane x = 2, the section becomes the curve z = f(2, y), which is a function of y alone.
Elliptic paraboloid
A surface shaped like a bowl, opening up or down. All cross-sections parallel to the base are ellipses. The quadratic form has all positive (or all negative) eigenvalues, with no cross-term large enough to change the sign pattern.
Hyperbolic paraboloid (saddle)
A surface shaped like a saddle or a Pringle crisp. It curves up in one direction and down in another. The quadratic form has eigenvalues of mixed sign.
Quadratic form
An expression of the type Ax² + Bxy + Cy². Its classification (elliptic, hyperbolic, or degenerate) depends on the discriminant AC − (B/2)², or equivalently on the eigenvalues of the associated matrix.
For T(x, y, z) = 2x + y + 3z + 27, the level sets are defined by 2x + y + 3z + 27 = c, which rearranges to 2x + y + 3z = c − 27.
Each value of c gives a different plane. These planes are all parallel (same normal vector ⟨2, 1, 3⟩) and are spaced apart by different constant values.
In geometric language: the level sets are a family of parallel planes with normal vector ⟨2, 1, 3⟩.
The bug at P(0, 1, 2) wants the 25°C level set: 2x + y + 3z + 27 = 25, or 2x + y + 3z = −2, or 2x + y + 3z + 2 = 0.
Current temperature at P: T(0, 1, 2) = 0 + 1 + 6 + 27 = 34°C.
Distance from (0, 1, 2) to the plane 2x + y + 3z + 2 = 0: d = |2(0) + 1(1) + 3(2) + 2| / √(4 + 1 + 9) = |0 + 1 + 6 + 2| / √14 = 9/√14.
The nearest point Q is found by moving from P in the direction of the normal ⟨2, 1, 3⟩ (or its negative) by the distance d. Since T(P) = 34 > 25, we move in the direction of decreasing T, which is −⟨2, 1, 3⟩.
Q = P − (9/14)⟨2, 1, 3⟩ = (0 − 18/14, 1 − 9/14, 2 − 27/14) = (−9/7, 5/14, 1/14).
The tangent plane at (a, b, f(a, b)) is: z − f(a, b) = f_x(a, b)(x − a) + f_y(a, b)(y − b).
Example: f(2, 5) = 7, f_x(2, 5) = 3, f_y(2, 5) = −2. Tangent plane: z − 7 = 3(x − 2) − 2(y − 5), which simplifies to z = 3x − 2y + 11.
If you slice z = f(x, y) with the plane x = 2, the resulting curve C lies in the plane x = 2 and is given by z = f(2, y).
The tangent line to C at (2, 5, 7) has direction vector ⟨0, 1, f_y(2, 5)⟩ = ⟨0, 1, −2⟩ (x is constant, y is the free parameter, z changes at rate f_y).
Parametric form: r(t) = ⟨2, 5 + t, 7 − 2t⟩.
Write the function as f(x, y) = Ax² + Bxy + Cy². The associated matrix is [[A, B/2], [B/2, C]].
If both eigenvalues are positive: elliptic paraboloid (bowl opening up).
If both eigenvalues are negative: elliptic paraboloid (bowl opening down).
If eigenvalues have mixed signs: hyperbolic paraboloid (saddle).
Quick test: compute AC − (B/2)². If positive, elliptic; if negative, hyperbolic.
f(x, y) = x² + 6xy + y²: A = 1, B = 6, C = 1. Discriminant: 1×1 − 9 = −8 < 0. Saddle.
g(x, y) = −4x² − y² + 4xy: A = −4, B = 4, C = −1. Discriminant: (−4)(−1) − 4 = 0. Degenerate (inverted trough).
h(x, y) = 7x² + 2xy + 4y²: A = 7, B = 2, C = 4. Discriminant: 28 − 1 = 27 > 0, and A > 0. Upward paraboloid.
Matching to figures: h is the upward paraboloid (Figure 1), f is the saddle (Figure 2), g is the inverted trough (Figure 3). Figure 4 (downward paraboloid) has no matching function.
z - f(a,b) = f_x(a,b)(x-a) + f_y(a,b)(y-b)d = \frac{|Aa + Bb + Cc + D|}{\sqrt{A^2 + B^2 + C^2}}\text{Discriminant of } Ax^2 + Bxy + Cy^2 = AC - \left(\frac{B}{2}\right)^2Students often write the tangent plane equation with the wrong sign on the f_y term. Double-check: z − f(a, b) = f_x(x − a) + f_y(y − b). There is a plus sign before f_y; the negative sign comes only if f_y itself is negative.
When finding distance to a level set, students sometimes plug in the wrong constant. You need the plane equation in the form Ax + By + Cz + D = 0 (with everything on one side) before applying the formula.
Students confuse the interior angle of the equilateral triangle (60°) with the direction of the normal vector to a plane. The normal vector to a level set of T(x, y, z) = 2x + y + 3z + 27 is the gradient ⟨2, 1, 3⟩, read directly from the coefficients.
For the graph-matching question, students sometimes try to plot points. The faster approach is computing the discriminant AC − (B/2)² to classify the surface.
⚠️ The tangent plane formula is one of the most-tested items in this course. Memorise it and practise applying it quickly.
⚠️ Point-to-plane distance uses the formula from the formula sheet. Make sure you can rearrange a level-set equation into standard form Ax + By + Cz + D = 0.
⚠️ Graph-matching problems test whether you can classify quadratic forms without graphing software. The discriminant test is the fastest route.
⚠️ When finding the tangent line to a section curve, remember that the fixed variable contributes 0 to the direction vector. If x = 2 is fixed, the direction is ⟨0, 1, f_y⟩, not ⟨1, 1, f_y⟩.
True or false: the level sets of a linear function in three variables are planes. (True.)
Fill in the blank: the normal vector to the plane 2x + y + 3z = 5 is ____. (⟨2, 1, 3⟩.)
True or false: a quadratic form with discriminant AC − (B/2)² > 0 gives a saddle surface. (False: it gives an elliptic paraboloid.)
Fill in the blank: the tangent plane to z = f(x, y) at (a, b) uses the partial derivatives ____ and ____. (f_x(a, b) and f_y(a, b).)
Q: The temperature in space is T(x, y, z) = 2x + y + 3z + 27. Describe the level sets geometrically and give their equations.
A: The level sets are parallel planes with normal vector ⟨2, 1, 3⟩. For temperature = c, the equation is 2x + y + 3z = c − 27.
Q: A bug is at P(0, 1, 2). How far is P from the nearest point where T = 25°C?
A: The plane is 2x + y + 3z + 2 = 0. Distance = |0 + 1 + 6 + 2|/√14 = 9/√14.
Q: Find the coordinates of that nearest point Q.
A: Q = (0, 1, 2) − (9/14)⟨2, 1, 3⟩ = (−9/7, 5/14, 1/14).
Q: Given f(2, 5) = 7, f_x(2, 5) = 3, f_y(2, 5) = −2, find the tangent plane to z = f(x, y) at (2, 5, 7).
A: z − 7 = 3(x − 2) − 2(y − 5), or z = 3x − 2y + 11.
Q: Find a parametric equation for the tangent line to the section curve at (2, 5, 7) when x = 2.
A: r(t) = ⟨2, 5 + t, 7 − 2t⟩.
Q: Classify f(x, y) = x² + 6xy + y². Is it a paraboloid or a saddle?
A: AC − (B/2)² = 1(1) − 9 = −8 < 0. Saddle (hyperbolic paraboloid).
Tangent planes lead into linear approximation and differentials. The gradient vector (whose components are the partial derivatives) is normal to level surfaces, connecting this material to directional derivatives later in the course. Quadratic-form classification foreshadows the second-derivative test for local extrema.
Related Terms / Search Tags: level set, level curve, level surface, contour, tangent plane, linear approximation, partial derivative, cross-section, section curve, elliptic paraboloid, hyperbolic paraboloid, saddle surface, quadratic form, discriminant, point-to-plane distance, normal vector, gradient, MATH 215, Calculus III, University of Michigan, multivariable calculus