12.5 Planes and Spheres in Space
1. Planes in R3 : The vector equation of a plane with direction vectors a and b through the point
P0 (x0 , y0 , z0 ) is given by
where x = x, y, z and p0 = x0 , y0 , z0 .
2. Denition of Normal: A vector n is normal to a pl
14.5 The Gradient Vector and Directional Derivatives
1. Gradient Vector: If f is a function of two variables x and y , then the gradient of f is the vector
function f dened by
2. Find the gradient of f (x, y ) = sin x + exy at (0, 1).
3. Find
g (a, b) if
13.4 Curvature
1. Consider a vector-valued function r(t) = f (t), g (t), h(t) , where t represents time. Imagine an
object moving along the curve C traced out by the endpoint of this function. Recall that the arc
length s(t) of the curve from u = a to u =
13.3 Arc Length and Speed
1. Arc Length in R2 :
2. Arc Length in R3 :
3. Arc Length in R3 : For the curve traced out by the endpoint of the vector-valued function r(t) =
f (t), g (t), h(t) , where f, f , g, g , h, h are all continuous scalar functions for
13.5 Motion in Three-Space
1. Speed: v (t) = s (t) = r (t)
Velocity: v(t) = r (t)
Acceleration: a(t) = v (t) = r (t)
2. Find the velocity and acceleration functions for the position function r(t) = te2t , 2e2t , 3t2 .
3. Find the velocity and position fun
14.1 Functions of Several Variables
1. A function of two variables is a rule that assigns to each ordered pair of real numbers (x, y ) in
a set D a unique real number denoted by f (x, y ).
2. The set D is the domain of f and its range is the set of values
14.4 Tangent Planes and Linear Approximations
1. Tangent Plane:
2. Tangent Plane and Normal Line:
3. Find the tangent plane and the normal line to the elliptic paraboloid z = 2x2 + y 2 at the point
(1, 1, 3).
4. Find the tangent plane and the normal line
14.3 Partial Derivatives
1. Let f (x, y ) be a function of two variables, where y = b is xed. Then g (x) = f (x, b) is a
function of a single variable x. If g has a derivative at a, then we call it the partial derivative
of f with respect to x at (a, b) a
13.2 The Calculus of Vector-Valued Functions
1. Limits: For a vector-valued function r(t) = f (t), g (t), h(t) , the limit of r(t) as t approaches a is
given by
lim r(t) = lim f (t), g (t), h(t) = lim f (t), lim g (t), lim h(t) ,
ta
ta
ta
ta
ta
provided a
13.1 Vector-Valued Functions
1. Vector-Valued Function: A vector-valued function r(t) is a mapping from its domain D R to
its range R R3 so that for each t in D we have r(t) = v for exactly one vector v R3 . A
vector-valued function can be written as
r(t)
12.2 Vectors in Space
1. We consider vectors in 3-dimensional Euclidean space R3 , usually consisting of three rectangular
coordinate axes x, y, z that intersect at the origin point O = (0, 0, 0).
2. Plotting (a, b, c) for a, b, c > 0: Move a units along
12.1 Vectors in the Plane
1. A vector is any quantity with a length (magnitude, norm) and a direction. We typically draw
an arrow to represent a vector. For example, let v be the vector from initial point A to terminal
point B . Then v = AB , and the magn
12.3 The Dot Product
1. Dot Product: The dot product of two vectors v = a1 , b1 and w = a2 , b2 in R2 is the scalar
vw =
Similarly, the dot product of two vectors v = a1 , b1 , c1 and w = a2 , b2 , c2 in R3 is
vw =
2. Compute the dot product v w for the f
12.6 Survey of Quadric Surfaces
Quadric Surface: The graph of the equation
Ax2 + By 2 + Cz 2 + Dxy + Eyz + F xz + ax + by + cz + d = 0
in R3 (three-dimensional space), where A, B, C, D, E, F, a, b, c, d are constants and at least one of
A, B, C, D, E, F i