Difficulty: Intermediate to Advanced | Prerequisites: Part 1 study notes (vectors, dot/cross products, lines and planes), single-variable derivatives and integrals, parametric equations.
This is the second half of a Calculus III course, covering how surfaces are described algebraically, how curves move through 3D space, and how differentiation extends to functions of several variables. These topics build directly on the vector algebra from Part 1. If you are comfortable with dot products, cross products, and equations of planes, you are ready. The gradient, directional derivatives, and the chain rule for multivariable functions show up in nearly every applied field, from physics to machine learning.
TL;DR
Surfaces in space are classified by their equations (ellipsoids, hyperboloids, paraboloids, cones, cylinders). Space curves are described by vector functions, with velocity, acceleration, curvature, and torsion measuring how the curve moves and bends. The gradient vector generalises the derivative to multiple variables, pointing in the direction of steepest increase.
Quadric surface
A surface defined by a second-degree equation in x, y, and z. The family includes ellipsoids, hyperboloids, paraboloids, and cones. Think of these as the 3D equivalents of conic sections (ellipses, hyperbolas, parabolas).
Cylinder (in calculus)
A surface generated by moving a 2D curve along a line parallel to one axis. Its equation is missing one variable. In simple terms, take a 2D shape and extend it infinitely in the third direction.
Space curve
A curve traced by a vector function r(t) = (x(t), y(t), z(t)) as the parameter t varies. Think of it as the path of a moving particle.
Arc length
The total distance along a curve, computed by integrating the speed |r'(t)| over the parameter interval.
Curvature (kappa)
A measure of how sharply a curve bends at a given point. A straight line has curvature 0; a tight circle has high curvature.
Torsion (tau)
A measure of how much a curve twists out of a single plane. A curve lying flat in a plane has torsion 0.
TNB frame (Frenet-Serret frame)
Three mutually perpendicular unit vectors that travel along a curve: the tangent T, the normal N (pointing towards the centre of curvature), and the binormal B = T x N.
Gradient vector
For a function f(x, y, z), the gradient is the vector of partial derivatives: nabla f = (df/dx, df/dy, df/dz). It points in the direction of the greatest rate of increase of f.
Directional derivative
The rate of change of f in a specified direction u. Computed as nabla f . u (the dot product of the gradient with the unit direction vector).
Partial derivative
The derivative of a multivariable function with respect to one variable, treating all others as constants. In simple terms, it is the slope along one axis while everything else is held fixed.
Differentiable (multivariable)
A function f(x, y) is differentiable at a point if it can be well approximated by a tangent plane there, with the error vanishing faster than the distance from the point.
Clairaut's theorem
If the mixed second partial derivatives f_xy and f_yx are both continuous, they are equal. In simple terms, the order of taking partial derivatives does not matter (under continuity).
Cylinders
A cylinder in Calc III is any surface whose equation is missing one variable. The missing variable is the axis along which the 2D cross-section is extruded.
Example: x² + y² = 4 is a circular cylinder (a tube) extending along the z-axis. z = x² is a parabolic cylinder extending along the y-axis.
Quadric surfaces
The six standard types, all defined by second-degree equations:
Ellipsoid: x²/a² + y²/b² + z²/c² = 1. All cross-sections are ellipses. When a = b = c, it is a sphere.
Hyperboloid of one sheet: x²/a² + y²/b² - z²/c² = 1. Connected surface, saddle-shaped cross-sections in some directions. Looks like a cooling tower.
Hyperboloid of two sheets: -x²/a² - y²/b² + z²/c² = 1. Two separate pieces. Cross-sections parallel to xy-plane are ellipses, but only exist when |z| > c.
Elliptic paraboloid: z = x²/a² + y²/b². Bowl-shaped, opening upward (or downward if negated). Cross-sections parallel to xy-plane are ellipses.
Hyperbolic paraboloid: z = x²/a² - y²/b². Saddle-shaped (a Pringle crisp). Cross-sections in one direction are parabolas opening up, in the other direction parabolas opening down.
Cone: x²/a² + y²/b² = z²/c². Two cones meeting at the origin. Cross-sections parallel to xy-plane are ellipses.
To identify the surface: look at the signs and powers of each variable, and take cross-sections by setting one variable to a constant.
A vector function r(t) = (x(t), y(t), z(t)) traces a curve in space as t varies over an interval.
Velocity: v(t) = r'(t), the derivative of the position vector. Its direction is tangent to the curve; its magnitude is the speed.
Acceleration: a(t) = r''(t), the derivative of velocity.
Arc length: the distance along the curve from t = a to t = b is L = integral from a to b of |r'(t)| dt.
Arc length parametrisation: re-parametrising a curve by arc length s (so that the parameter equals distance travelled) is useful theoretically because |r'(s)| = 1 everywhere.
Curvature (kappa) measures how fast the curve is changing direction at a point.
Formula: kappa = |r'(t) x r''(t)| / |r'(t)|³.
For a circle of radius R, kappa = 1/R. Smaller radius means sharper bend.
A straight line has kappa = 0 everywhere.
Torsion (tau) measures how much the curve twists out of the osculating plane (the plane that best fits the curve at that point).
Formula: tau = (r'(t) x r''(t)) . r'''(t) / |r'(t) x r''(t)|².
A curve confined to a single plane has tau = 0.
Positive torsion means the curve twists like a right-handed helix; negative like a left-handed one.
Tangent vector: T(t) = r'(t) / |r'(t)|. The unit vector in the direction of motion.
Normal vector: N(t) = T'(t) / |T'(t)|. Points towards the centre of the curve's curvature (the direction the curve is turning).
Binormal vector: B(t) = T(t) x N(t). Perpendicular to both T and N, completing a right-handed coordinate frame.
Together, T, N, B form the Frenet-Serret frame (TNB frame), a moving coordinate system that rides along the curve.
The osculating plane is spanned by T and N. The curve lies closest to this plane near the given point.
Partial derivatives: f_x = df/dx holds y (and z) constant, and vice versa. These are the building blocks of multivariable differentiation.
Gradient vector: nabla f = (f_x, f_y, f_z). The gradient points in the direction of steepest ascent of f, and its magnitude |nabla f| is the maximum rate of change.
Directional derivative: the rate of change of f in the direction of a unit vector u is D_u f = nabla f . u.
Maximum when u is parallel to nabla f (uphill).
Zero when u is perpendicular to nabla f (along a level curve/surface).
Minimum (most negative) when u is anti-parallel to nabla f (downhill).
The gradient is always perpendicular to the level curves (2D) or level surfaces (3D) of f. This is why contour lines on a topographic map are perpendicular to the steepest slope.
Linear approximation
Near a point (x_0, y_0), a differentiable function can be approximated by its tangent plane: f(x, y) is approximately f(x_0, y_0) + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0). This is the multivariable equivalent of the tangent-line approximation from Calc I.
Differentials
df is approximately f_x dx + f_y dy. This estimates the change in f from small changes dx and dy in the inputs. Useful for error estimation in applied problems.
Continuity
A function f(x, y) is continuous at (x_0, y_0) if the limit as (x, y) approaches (x_0, y_0) equals f(x_0, y_0). In multiple variables, the limit must be the same regardless of the path of approach.
Differentiability
A function is differentiable at a point if its linear approximation (tangent plane) exists and the error goes to zero faster than the distance from the point. A function can have partial derivatives at a point without being differentiable there.
Clairaut's theorem
If f_xy and f_yx are both continuous near a point, then f_xy = f_yx. This means you can take mixed partial derivatives in either order.
Multivariable chain rule
For a composite function, the chain rule involves sums of products of partial derivatives. For example, if z = f(x, y) where x = g(t) and y = h(t), then dz/dt = f_x * dx/dt + f_y * dy/dt.
Implicit differentiation
When a relationship F(x, y, z) = 0 defines z implicitly as a function of x and y, the partial derivatives can be found using dz/dx = -F_x / F_z and dz/dy = -F_y / F_z.
Differential equations in space (overview)
Three important PDEs that appear in applications: Laplace's equation (nabla² u = 0, potential fields), the wave equation (u_tt = a² nabla² u, vibrations), and the heat equation (u_t = k nabla² u, heat conduction). These are typically studied in depth in later courses but rely on the gradient and partial derivatives from this material.
Key formulas
Formula | Expression |
|---|---|
Curvature | kappa = |
Torsion | tau = (r' x r'') . r''' / |
Unit tangent | T = r' / |
Unit normal | N = T' / |
Binormal | B = T x N |
Arc length | L = integral of |
Gradient | nabla f = (f_x, f_y, f_z) |
Directional derivative | D_u f = nabla f . u |
Linear approximation | f(x,y) approx f(x0,y0) + f_x(x-x0) + f_y(y-y0) |
Differential | df approx f_x dx + f_y dy |
Implicit diff (z from F=0) | dz/dx = -F_x / F_z |
Real-world applications
The gradient vector is used in machine learning (gradient descent optimises a loss function by moving opposite to the gradient) and in physics (electric fields are the negative gradient of electric potential).
Curvature determines how roads and railways are banked. Civil engineers compute the radius of curvature to set safe speed limits on curves.
The TNB frame describes the orientation of an aircraft or spacecraft moving along a trajectory, relating directly to pitch, yaw, and roll.
Quadric surfaces model satellite dishes (paraboloids), nuclear cooling towers (hyperboloids of one sheet), and the Earth itself (an ellipsoid).
Students often think that having partial derivatives at a point guarantees differentiability. It does not. A function can have both f_x and f_y defined at a point while still failing to be differentiable there (the function may not even be continuous).
Students confuse the gradient with the directional derivative. The gradient is a vector; the directional derivative in a given direction is a scalar (the dot product of the gradient with the direction).
A common mistake with quadric surfaces is misidentifying the type by looking only at the variables present. You need to check the signs: x²/a² + y²/b² - z²/c² = 1 is a hyperboloid of one sheet, while -x²/a² - y²/b² + z²/c² = 1 is a hyperboloid of two sheets. The sign pattern matters.
Students sometimes assume the normal vector N points outward. It does not. N always points toward the centre of curvature, which may be inward.
⚠️ Identifying and sketching quadric surfaces is a standard exam topic. Practise classifying by setting one variable constant and examining the resulting cross-section.
⚠️ The curvature formula (especially the cross-product version) appears regularly on exams. Be fluent with the computation.
⚠️ Gradient and directional derivative problems are heavily tested. Know that the maximum rate of change equals |nabla f| and occurs in the direction of nabla f.
⚠️ Linear approximation and differentials are often tested as applied estimation problems ("estimate the change in f when x changes by 0.1 and y changes by -0.05").
⚠️ Clairaut's theorem is a common true/false or short-answer question. Know the continuity condition.
True or false: The gradient vector is always perpendicular to level curves/surfaces. (True.)
Fill in the blank: A curve lying entirely in a plane has torsion equal to ___. (0)
True or false: If f_x and f_y both exist at a point, then f is differentiable there. (False.)
True or false: The equation x² + z² = 9 (with y absent) describes a sphere. (False, it is a cylinder.)
Fill in the blank: The directional derivative is maximised when the direction is parallel to ___. (the gradient vector)
Q: Classify the surface x² + 4y² - z² = 4.
A: Divide through by 4: x²/4 + y² - z²/4 = 1. This has the form x²/a² + y²/b² - z²/c² = 1, so it is a hyperboloid of one sheet.
Q: For f(x, y) = x²y + 3xy², find the gradient at the point (1, 2).
A: f_x = 2xy + 3y² = 2(1)(2) + 3(4) = 16. f_y = x² + 6xy = 1 + 12 = 13. So nabla f(1, 2) = (16, 13).
Q: Find the directional derivative of f(x, y) = x²y + 3xy² at (1, 2) in the direction of v = (3, 4).
A: The unit vector is u = (3/5, 4/5). From above, nabla f = (16, 13). D_u f = 16(3/5) + 13(4/5) = 48/5 + 52/5 = 100/5 = 20.
Q: Use a linear approximation to estimate f(1.02, 1.97) given f(1, 2) = 10, f_x(1, 2) = 16, f_y(1, 2) = 13.
A: f(1.02, 1.97) is approximately 10 + 16(0.02) + 13(-0.03) = 10 + 0.32 - 0.39 = 9.93.
Q: Find the curvature of r(t) = (cos t, sin t, t) at any t.
A: r' = (-sin t, cos t, 1), |r'| = sqrt(2). r'' = (-cos t, -sin t, 0). r' x r'' = (sin t, -cos t, 1). |r' x r''| = sqrt(2). So kappa = sqrt(2) / (sqrt(2))³ = 1/2. The helix has constant curvature 1/2.
Connections to other topics
The gradient and directional derivatives connect forward to Lagrange multipliers and optimisation of multivariable functions. Curvature and the TNB frame connect to line integrals and the motion of charged particles in electromagnetic fields. Quadric surfaces reappear when you evaluate double and triple integrals, since the limits of integration are often defined by these surfaces. The chain rule for multivariable functions extends to change-of-variable techniques (Jacobians) used in multiple integration.
Related Terms / Search Tags
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