Differential Calculus of multivariable functions Flashcards
Real function of a single variable
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.
Function of several real variables
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.
Vertical sections in x-z plane
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
Vertical sections in y-z plane
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.
Level curves
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.
Partial derivative
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.
Sufficiently smooth (informal)
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.
The chain rule for functions of a single variable
Suppose that y = y(x) and x = x(u). We denote y = y(x(u)).
Then
dy/du = dy/dx * dx/du
Chain rule for functions with two variables
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
Key points of the chain rule
1) Write fs = fxxs+fyys + … + fz*zs
2) Calculate derivatives required
3) Sub the values back in and simplify
A vector
A vector is a quantity that involves both magnitude and direction
Magnitude of a vector
The magnitude of a vector v is the length of the arrow, denoted |v|.
Unit vector
Magnitude of 1
Dot product of two vectors
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
Cross product two vectors
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.
Perpendicular lines
Two lines a and b are perpendicular iff a.b = 0
Parallel lines
Two lines a and b are parallel iff axb =0
Plane
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
Tangent plane
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.
Curve
A curve C van be expressed in the parametric form
r = r(t) = x(t) i + y(t) j + z(t) k, a =
Tangent vector of a curve
T = dr/dt
= dx/dt i + dy/dt j + dz/dt k
Gradient vector
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.
Normal to the level curve theorem
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)
Normal to a level surface theorem
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)
Finding the tangent plane to a surface
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
Directional derivatives (Theory)
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.
Directional derivative (Wordy)
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.
Directional derivative equation
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|
Relationship between gradient and directional derivatives
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)
Du is linear
For any functions f and g and any scalar constants s and t
Du(sf + tg) = s Du(f) + t Du(g)
Steepest ascent theorem
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.
