Multivariable Calculus

Question 1

Answer 1

Question 2

Answer 2

Question 3

Answer 3

Question 4

Answer 4

Question 5

Answer 5

Question 6

Answer 6

Question 7

The dot product is a

Answer 7

Scalar

Question 8

Answer 8

0

Question 9

Answer 9

Question 10

Answer 10

Question 11

Answer 11

Question 12

What does a gradient tell you?

Answer 12

The gradient tell you how fast something is changing and in which direction–all at once. If you’ve got a function like f(x,y) that describes something like temperature on a map or elevation on a landscape, the gradient at a point is a vector that:
- points in the direction where the function increases the fastest; and
- has a magnitude equal to that maximum rate of increase.

Question 13

Answer 13

Question 14

Answer 14

Question 15

Answer 15

Question 16

Answer 16

Question 17

Question 18

The cross product of two vector is

Answer 18

a vector that is orthogonal to the original vectors (i.e., orthogonal to the plane determined by the original vectors)

Question 19

The angle between two planes is

Answer 19

the same as the angle between their normal vectors

Question 20

Answer 20

Question 21

Question 22

Question 23

Question 24

What does the gradient mean?

Answer 24

The gradient is a mathematical tool that tells you which direction to move in to increase a function the fastest.
- the gradient vector is orthogonal to level curves in the plane
- the gradient vector is orthogonal to level surfaces in space

Question 25

What does the dot product do?

Answer 25

The dot product is a way of combining two vectors that tell you how much they're pointing in the same direction: - pointing in the same direction = positive - perpendicular = 0 - pointing opposite = negative

Question 26

Question 27

Answer 27

Question 28

What is a directional derivative?

Answer 28

A directional tells you how a function changes as you move in a specific direction from a given point. For example, if you're standing on a hilly surface defined by a function f(x,y), and you want to know how steep the hill is as you walk northeast, the directional derivative is the answer.