Differential Calculus of multivariable functions

Question 1

Real function of a single variable

Answer 1

y = f(x)

where x e D, y e C.
(D is the domain of f and C is the range of f)

y depends on the single real variable x.

Question 2

Function of several real variables

Answer 2

Function that depends on more than one real variable.

A function f of n real variables is a rule that assigns a unique real number f( x1, x2, … , xn) to each point (x1, …, xn)

where the domain of f is a subset of |R^n and the range of f is a subset of |R.

Question 3

Vertical sections in x-z plane

Answer 3

For a function z = f(x , y), a vertical section z= f(x , c) is the intersection of the vertical plane y=c and the surface z=f(x , y), where c is a constant.

This will produce a graph in the x-z plane

Question 4

Vertical sections in y-z plane

Answer 4

For a function z = f(x , y), a vertical section z = f(c, y) is the intersection of the vertical plane x=c and the surface z = f(x , y), where c is a constant.

This will produce a graph in the y-z plane.

Question 5

Level curves

Answer 5

For a function z=f(x , y) a level curve c= f(x,y) is the intersection between the horizontal plane z=c and the surface z= f(x, y), where c is a constant

This will produce a graph in the x-y plane.

Question 6

Partial derivative

Answer 6

A function f : |R^3 -> |R is said to have a partial derivative with respect to x at the point (x0, y0, zo) if the following limits exists

lim (h->0)[ f(xo + h, yo, zo) - f(xo, yo, zo) ] / h.

This limit is the partial derivative of f w.r.t x at (xo, yo, z0)

To calculate this treat all other variables as constants.

Partial derivatives w.r.t x and y are calculated in the same way.

Question 7

Sufficiently smooth (informal)

Answer 7

A function is said to be sufficiently smooth if the function and as many partial derivatives as required are continuous where they need to be.

Question 8

The chain rule for functions of a single variable

Answer 8

Suppose that y = y(x) and x = x(u). We denote y = y(x(u)).
Then
dy/du = dy/dx * dx/du

Question 9

Chain rule for functions with two variables

Answer 9

Consider f(x , y). Where x = x(u, v) and y = y(u, v).

df/du = df/dx * dx/du + df/dy * dy/du
df/dv = df/dx * dx/dv + df/dy * dy/dv

Question 10

Key points of the chain rule

Answer 10

1) Write fs = fxxs+fyys + … + fz*zs
2) Calculate derivatives required
3) Sub the values back in and simplify

Question 11

A vector

Answer 11

A vector is a quantity that involves both magnitude and direction

Question 12

Magnitude of a vector

Answer 12

The magnitude of a vector v is the length of the arrow, denoted |v|.

Question 13

Unit vector

Answer 13

Magnitude of 1

Question 14

Dot product of two vectors

Answer 14

Consider two vectors
a= a1i + a2j + a3k and b = b1i + b2j + b3k.

The dot product of the two vectors is
a.b = |a||b|cos& = a1b1 +a2b2 + a3b3

Question 15

Cross product two vectors

Answer 15

a x b = |a| |b| sin&n = (a2b3 - a3b2)i + (a3b1 - a1b3) j + (a1b2 - a2b1) k.

where n is the unit normal to both a and b such that (a, b, n) forms a right handed system.

Question 16

Perpendicular lines

Answer 16

Two lines a and b are perpendicular iff a.b = 0

Question 17

Parallel lines

Answer 17

Two lines a and b are parallel iff axb =0

Question 18

Plane

Answer 18

The vector equation of a plane is r.n = a.n

where a is a point in the plane, n is a normal vector to the plane and r a general point in the plane

Question 19

Tangent plane

Answer 19

If n is a normal vector to the surface S at the point a, then the vector equation of the tangent plane to S at A is given by r.n=a.n.

Since a is at the tangent plane and the normal of the tangent plane is the same as the normal of the surface at a.

Question 20

Curve

Answer 20

A curve C van be expressed in the parametric form

r = r(t) = x(t) i + y(t) j + z(t) k, a =

Question 21

Tangent vector of a curve

Answer 21

T = dr/dt

= dx/dt i + dy/dt j + dz/dt k

Question 22

Gradient vector

Answer 22

Let f: |R^3 -> |R be a function of three variables x , y , z.
If the partial derivatices of f with respect to x, y and z exist at the point (a , b ,c), then the gradient of f at (a, b, c) is defined to be the vector

▽f(a,b,c) = gradf(a,b,c) = ∂f(a,b,c)/∂x i + ∂f(a,b,c)/∂y j + ∂f(a,b,c)/ ∂z k

A gradient vector of an n-variable function is a vector function in n dimensions.

Question 23

Normal to the level curve theorem

Answer 23

Let f: |R^2 -> |R be a sufficiently smooth function and let (a,b) be a point on the level curve f(x,y) =k for some constant k.
If ▽f(a,b) ≠ 0, then the vector ▽f(a,b) is normal to the level curve at f(x, y) =k at the point (a,b)

Question 24

Normal to a level surface theorem

Answer 24

Let f: |R^3 -> |R be a sufficiently smooth function and let (a,b,c) be a point on the level curve f(x,y,z) =k for some constant k.
If ▽f(a,b,c) ≠ 0, then the vector ▽f(a,b,c) is normal to the level curve at f(x, y) =k at the point (a,b,c)

Question 25

Finding the tangent plane to a surface

Answer 25

Let the surface be given by f(x, y, z) = k a the point (a,b,c).
1) Write the surface in the form of a level surface
f(x , y , z) = k
2) Calculate the normal vector to the surface at the point a n= ▽f(a)
3) The tangent plane is given by r.n=a.n

Question 26

Directional derivatives (Theory)

Answer 26

Tells us how fast f(x, y) changes value as we move through the domain of f at (a,b) in some other direction.
We can specify the direction using a non-zero vector e.g the unit vector.

Question 27

Directional derivative (Wordy)

Answer 27

Let u = u1 i + u2 j + u3 k be a unit vector. The directional derivative of f(x , y , z) at the point A = (a , b , c) in the direction of u is the rate of change of f(x , y , z) w.r.t distance measured at A along the ray in the direction of u in the x-y-z field.
Denote the distance |AP| = t, where P is at r = a + tu.

Question 28

Directional derivative equation

Answer 28

The direction derivative is given by 
Du f(a,b,c) = lim ( t -> 0) [f(a+tu1, b+tu2, c+tu3) - f(a,b,c) ] / t
Du f(a,b,c) = lim ( t -> 0) [ f(a+tu) - f(a)] / t
Du f(a,b,c) = lim (P -> A) [f(P) - f(A)] / |AP|

Question 29

Relationship between gradient and directional derivatives

Answer 29

Let f: |R^3->|R be sufficiently smooth and u = u1 i + u2 j + u3 k be a unit vector in |R^3. Then 
Du f(a,b,c) = u . ▽f(a,b,c)

Question 30

Du is linear

Answer 30

For any functions f and g and any scalar constants s and t

Du(sf + tg) = s Du(f) + t Du(g)

Question 31

Steepest ascent theorem

Answer 31

Let f be a function of several real variables. Suppose that ▽f exists and is non-zero at a point a.
Then ▽f(a) is the direction of steepest ascent of the function f at the point a and -▽f(a) is the direction of the steepest descent.