Calculus 3

Front 1

What is the distance formula in 3D?

Back 1

Front 2

What is the formula for a sphere in 3D?

Back 2

Front 3

What is a scalar?

Back 3

Is a quantity that has a magnitude describable by a real number and no direction.

Front 4

What is a vector?

Back 4

Is used to indicate a quanitity (such as displacement or velocity) that has both magnitude and direction.

Front 5

Dot Product

Back 5

Front 6

Scalar projection of b onto a

Back 6

Front 7

Scalar projection of a onto b

Back 7

Front 8

Vector projection of b onto a

Back 8

Front 9

Vector projection of a onto b

Back 9

Front 10

Angle between vectors a and b of a dot product.

Back 10

Front 11

When are vectors a and b called perpendicular or orthogonal?

Back 11

When the angle between them makes cosine zero such angles 0 deg, 180 deg, 360 deg, etc.

Two vectors a and b are orthogonal if and only if a dot b = 0.

Front 12

How to find the angle of a vector product?

Back 12

Front 13

When are two non-zero vectors a and b parallel?

Back 13

If and only if a x b = 0

Front 14

What are the names of the angles between vector v?

Back 14

The angle between vector v is:
- x-axis is alpha
- y-axis is beta
- z-axis is gamma

Front 15

What is the graph of a Line?

Back 15

Front 16

What is the vector equation of a line?

Back 16

- r: The equation of the line.

- r =

- r0 =

- v =

- t = a parameter

Front 17

Equation of a Line

Back 17

Each t defines a point on the line.

Front 18

Unit Tangent Vector

Back 18

Front 19

Magnitude (Length) of a Vector

Back 19

Front 20

Ellipsoid Surface

Back 20

Front 21

Cone Surface

Back 21

Front 22

Elliptic Paraboloid Surface

Back 22

Front 23

Hyperboloid of One Sheet Surface

Back 23

Front 24

Hyperbolic Paraboloid Surface

Back 24

Front 25

Hyperbolic of Two Sheets Surface

Back 25

Front 26

Example of Vector Spaces

Back 26

Front 27

What is a basis?

Back 27

A basis for a vector space is a set of vectors belonging to the vector space and that are linearly independent and that spans the vector space.

Front 28

What is a spanning set?

Back 28

A set of vectors is a spanning set of a vector space if every vector from the vector space can be written as a linear combination of the vectors from the spanning set.

Front 29

Equation of a plane

Back 29

Front 30

Formula for the distance between a point and a plane

Back 30

Front 31

Unit Normal Vector

Back 31

Front 32

Unit Binormal Vector

Back 32

Front 33

Curvature of a Curve

Back 33

Front 34

Arc Length of a Curve

Back 34

Front 35

Normal Plane

Back 35

Is a plane containing N(t) and B(t) and has T(t) as its normal vector.

Front 36

Osculating Plane

Back 36

Is a plane containing T(t) and N(t) and has B(t) as its normal vector. N(t) points to the center of the osculating circle.

Front 37

Rectifying Plane

Back 37

Is a plane containing T(t) and B(t) and has N(t) as its normal vector.

Front 38

Equation of a Tangent Plane for w=f(x,y,z)

Back 38

Front 39

Change in f in the direction of vector u

Back 39

Front 40

Maximum change in f

Back 40

Front 41

Chain rule for the derivative of F with respect to s

Back 41

Front 42

Linear Approximation Equation

Back 42

Front 43

Equation of a Tangent Plane for z=f(x,y)

Back 43

Front 44

Volume of Parallelepiped

Back 44

Front 45

Relative Minimum of a function with two variables

Back 45

Front 46

Relative Maximum of a function with two variables

Back 46

Front 47

Critical points of a function with two variables

Back 47

Front 48

Second Derivative Test of a function with two variables

Back 48