Calc III Conceptual/Definitive Terms

What do the coefficients A, B, C represent in a plane equation Ax + By + Cz = D?

They represent the components of the normal vector to the plane.

When are two planes parallel?

When their normal vectors are scalar multiples of each other

What do you need to define a line in 3D?

A point and a direction vector

What do you need to define a plane?

Either: A point + normal vector, or three points to form two direction vectors

What does the cross product give you geometrically?

A vector perpendicular to both original vectors

What does the dot product tell you?

The angle between two vectors and whether they are perpendicular (dot = 0)

What does r(t)=x(t)i+y(t)j+z(t)k represent?

The position vector of a moving object in 3D

What does x(t), y(t), and z(t) individually represent?

Position in each coordinate direction

What is velocity in terms of r(t)?

v(t) = r(t)

what is acceleration in terms of r(t)?

a(t) = r’’(t)

What is speed?

The magnitude of velocity: |v(t)|

When is a function NOT smooth?

When r’(t) = 0 ( zero velocity vector)

What does it mean if you get a 0/0 when evaluating a limit?

The limit may or may not exist—more work must be done

Can you use L’Hopital’s Rule for multivariable limits?

No

When should you use polar coordinates in limits?

When approaching (0,0) and expressions that involve x² + y²

How do you prove a limit does NOT exist?

Use the method of paths (different paths —> different results)

What does f_x represent?

Rate of change (slope) in the x direction

What does f_y represent?

Rate of change in the y direction

What is the gradient vector ∇f?

A vector pointing in the direction of maximum increase

What is special about the gradient vector geometrically?

it is perpendicular (normal) to level curves/surfaces

What does a directional derivative represent?

The rate of change of a function in a specific direction

What does a tangent plane represent?

A linear approximation of a surface at a point.

What determines a normal line?

The gradient vector (it gives the direction).

What are critical points?

Points where f_x = 0 and f_y = 0

What are the possible types of critical points?

Relative minimum, relative maximum, saddle point

What is a saddle point?

A point that is not a max or min, but changes direction

When do you use Lagrange multipliers?

When optimizing with a constraint

What is the key equation in Lagrange multipliers?

∇ f = λ∇g

Why do Lagrange extrema differ from regular extrema?

Because they must satisfy a constraint

What are absolute extremas?

The largest and smallest values overall, including boundaries. (Kind of like candidates test, where you see the absolute max/mins based on critical points and boundaries(end pts))

What can double integrals represent?

Area, volume, mass, etc

Why change order of integration?

To make the integral easier to evaluate.

When do you switch to cylindrical coordinates?

When the region involves circles or cylinders.

What does a line integral represent physically?

Work done by a force field.

When can you use the Fundamental Theorem of Line Integrals?

When the field is conservative

What is Green’s Theorem used for?

Converting a line integral → double integral.

What is flux?

The amount of a field flowing through a surface

What does the gradient tell you about extrema?

It points in the direction of steepest increase, so zero gradient → possible extrema.

Why is the gradient perpendicular to level curves?

Because there is no change along the curve, so the direction of change must be perpendicular.

What is the difference between flux and circulation?

• Flux: flow through a surface

• Circulation: rotation along a boundary