Differential Calculus

Question 1

Partial Differnetiation

Answer 1

Partial differentiation can be used to determine the rate of change along either x and y direction.

These are denoted by ∂f/∂x ∂f/∂y
where ∂ denotes a partial rather than absolute derivative

Question 2

2nd order Partial derivatrives

Answer 2

∂f/∂x → ∂^2f/∂x^2 = ∂/∂x(∂f/∂x)
→ ∂^2f/∂y.∂x = ∂/∂y(∂f/∂x)
∂f/∂y → ∂^2f/∂y^2 = ∂/∂y (∂f/∂y)
→ ∂^2f/∂x.∂y = ∂/∂x(∂f/∂y)

∂^2f/∂x∂y = ∂^2f/∂y∂x

Question 3

Multivariable Chain rule

Answer 3

dw/dt = dw/dx.dx/dt

For a multivariable function w(x,y,z ) where each independent variable depends on one variable, t , the chain rule becomes
dw/dt = ∂w/∂x. dx/dt + ∂w/∂y.dy/dt +∂w/∂z . dz/dt

w(x(t),y(t),z(t))

Question 4

Independent variable depend on 2variables

Answer 4

w(x(u,v), y(u,v),z(u,v))
∂w/∂u = ∂w/∂x .∂x/∂u + ∂w/∂y.∂y/∂u + ∂w/∂z.∂z/∂u

∂w/∂v = ∂w/∂x .∂x/∂v + ∂w/∂y. ∂y/∂v + ∂w/∂z.∂z/∂v

Question 5

Stationary Points

Answer 5

For function f(x) , sp exists where df/dx = 0
The 2nd order derivation can be used to classify the point d^2f/dx^2> 0 local min
point d^2f/dy^2 < 0 local max
point d^2f/dx^2 = 0 local min, max or inflection point

Question 6

Stationary points of two variable f

Answer 6

∂f/∂x = 0
∂f/∂y = 0
these two conditions must satisfy simultaneously
For f(x,y) there are 3 types of SP
local min
local max
saddle point - f value increase along one direction and decreases along the other direction

Question 7

Hessian matrix

Answer 7

can be used in the classifcation of SP
For a function f(x,y) , the Hessian matrix is given by:
H = (2,2) ( ∂^2f/∂x^2, ∂^2f/∂x.∂y, ∂^2f/∂y.∂x, ∂^2f/∂y^2)

Determinant (det(H))
= (∂^2f/∂x^2 * ∂^2f/∂y^2 ) - (∂^2/∂x.∂y * ∂^2f/∂y.∂x)

Question 8

Classification from hessian

Answer 8

Classifcation det(H) ∂^2f/∂x^2 or ∂^2f/∂y^2
Local min >0 >0
Local max >0 <0
Saddle point <0
Inconclusive =0

Question 9

Vector calculus

Answer 9

three quantities related to scalar and vector fields
gradient
divergence
curl

Question 10

Del Operator

Answer 10

Calculating the gradient, divergence or culr requires the use of del operation .

nabla = (∂/∂x)i + (∂/∂y)j +(∂/∂z) k

Summary of output of nabla
Property Operation Input Function Output Function
Gradient nabla f Scalar Vector
Divergence nabla.vector(v) Vector Scalar
Curl ∂ * vector(v) Vector Vector

Question 11

Gradient

Answer 11

The gradient of a scalar function can be found using nabla f(x,y,z ) = (∂f/∂x)i + (∂f/∂y)j + (∂f/∂z)k

The gradient of a scalar field is itself a vector function that has two informative properties :
→ it points in the direction of maximum rate of change of f
→ its magnitude gives the maximum rate of change of f

Directional derivative:
Th rate of change of f in a direction vector(b) can be calculated by calculating the overlap in magnitude with the gradient of f:
Dbf = unit vector b . nabla(f) = b/ II b II . nabla(f)

where Dbf = directional derivative

Question 12

Divergence

Answer 12

The dot product of del and a vector function gives a property known as divergence
nabla . vector(v(x,y,z)) = (∂/∂x, ∂/∂y, ∂/∂z ) . (vx,vy,vz)

Divergence is a scalar function that represents the local expansion or contraction of the vector function

For example if a vector function represents fluid velocity then
→ The fluid is locally expanding where divergence >0
→The fluid is locally contracting where divergence < 0
→ The fluid is incompressible if divergence = 0 , everywhere In other words the inflow and outflow are balanced locally.

Question 13

Curl

Answer 13

The cross product of del and a vector function gives a property known as the curl

curl is a vector function that indicates the direction and magnitude of the loca rotation of the vector function
The curl vector points along the axis of rotation with the right hand rule determining whether the direction of rotation.

If curl is zero everywhere, the vector function is irrotational

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