Math: Multivariable Calculus Basics

What is the formula for the distance between two points P(x1, y1, z1) and Q(x2, y2, z2) in 3D space?

The distance is the square root of ((x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2).

What is the geometric interpretation of the dot product of two non-zero vectors u and v?

The dot product u · v equals |u||v|cos(θ), where θ is the angle between them. It measures how much one vector points in the direction of the other.

How do you find a vector that is orthogonal to two given non-parallel vectors u and v in 3D space?

You compute the cross product u × v, which yields a vector orthogonal to both u and v.

How do you find the unit tangent vector T(t) for a smooth parametric curve r(t)?

The unit tangent vector is found by dividing the velocity vector by its magnitude: T(t) = r'(t) / |r'(t)|.

What does the partial derivative f_x(x,y) represent geometrically?

The partial derivative f_x(x,y) represents the slope of the tangent line to the surface z = f(x,y) in the direction of the positive x-axis, treating y as a constant.

What is the gradient vector ∇f(x,y) and what is its primary geometric property?

The gradient vector ∇f(x,y) = ⟨f_x, f_y⟩ points in the direction of the maximum rate of increase of the function f at that point.

What is the formula for the directional derivative of f(x,y) in the direction of a unit vector u?

The directional derivative is D_u f(x,y) = ∇f(x,y) · u, which is the dot product of the gradient and the unit vector.

What is the equation of the tangent plane to the surface z = f(x,y) at the point (x0, y0, z0)?

The equation of the tangent plane is z - z0 = f_x(x0, y0)(x - x0) + f_y(x0, y0)(y - y0).

How do you locate the critical points of a differentiable function of two variables, f(x,y)?

Critical points are located where both partial derivatives are zero (∇f = 0) or where the gradient is undefined.

In the Second Derivative Test for f(x,y), what does it mean if the discriminant D = (f_xx)(f_yy) - (f_xy)^2 < 0 at a critical point?

If D < 0 at a critical point, the point is a saddle point, meaning it is neither a local maximum nor a local minimum.