Vectors, Planes, Quadratic Surfaces
This is the equation of a plane passing through the origin with normal vector ⟨2,3,−1⟩
What is 2x+3y−z=0
The derivative of r(t) = ⟨t2,sin(t),et⟩
What is ⟨2t,cos(t),et⟩
The partial derivative of f(x,y)=x2y+y3 with respect to x.
What is 2xy?
The gradient of f(x,y)=x2+y2 at the point (1,2).
What is ⟨2,4⟩.
The linearization L(x,y) of f(x,y) at a point (a,b) is this expression.
What is f(a,b)+fx(a,b)(x−a)+fy(a,b)(y−b)?
This quadric surface is described by
(x2/4)+(y2/9)+(z2/16) = 1
What is an ellipsoid?
The arc length of r(t)=⟨cost,sint,t⟩ from t=0 to t=2π.
This theorem allows you to switch the order of partial differentiation for well-behaved functions: fxy = fyx
What is Clairaut's theorem?
The gradient vector at a point on a surface is always perpendicular to this.
What is the level curve (contour) through that point?
This coordinate system replaces (x,y,z) with (r,θ,z).
What are cylindrical coordinates?
The distance from the point (1,2,3) to the plane 2x−y+2z=5
What is 1/3?
The arc length of r(t) = ⟨3t,4t⟩ from t = 0 to t = 1.
What is 5?
Evaluate ∫(0,1)∫(0,1) of x2y dydx
What is 1/4?
At a local maximum or minimum of f(x,y) these must both equal zero.
What are fx and fy?
The directional derivative of f in the direction of unit vector u is given by this formula.
What is Du(f)=∇f⋅u?
This is the name of the surface described by z = x2- y2
What is a hyperbolic paraboloid (saddle)?
For a space curve, this vector points toward the center of curvature and is perpendicular to T.
What is the principal unit normal or n?
The chain rule expression for dz/dt where z = f(x,y) x = x(t) and y = y(t)?
What is (∂f/∂x)(dx/dt) + (∂f/∂y)(dy/dt)?
This second derivative test quantity D=fxxfyy−fxy2 being negative at a critical point indicates this type of point.
What is a saddle point?
The Jacobian of the transformation x=rcosθ y=rsinθ from polar to Cartesian.
What is r?
Find the equation of the plane containing the points (1,0,0), (0,1,0), and (0,0,1).
What is x+y+z=1?
The unit tangent vector T and the equation of the tangent line to r(t)=⟨t2,t3,t⟩ at t=1.
T(1) = 1/sqrt(14)⟨2,3,1⟩ and l(t) = ⟨1+2t,1+3t,1+t⟩
Convert ∫(0,1)∫(0,sqrt(1-x2)) of (x2+y2)dydx to polar and evaluate.
What is pi/8?
Use Lagrange multipliers to find the maximum of f(x,y)=xy subject to x+y=10.
What is 25, at x=y=5?
This theorem relates a double integral over a region D to a line integral over its boundary C: ∮CP dx+Q dy=∬D(∂Q/∂x−∂P/∂y)dA.
What is green's theorem?