Complete Theory

Worked Examples

Example 1Jacobian and local invertibility

Example 2Area of the unit sphere

Example 3Lagrange: maximum under a constraint

Example 4Jacobian determinant of polar coordinates

Example 5Implicit function: a system

Exercise 1JacobianMedium

📋 Problem to solve

Compute

J_f(0,0)

for

f(x,y)=(\sin x+y,\,e^y-x)

Exercise 2LagrangeHard

📋 Problem to solve

Maximum of

f=x+y+z

x^2+y^2+z^2=1

Exercise 3Inverse functionMedium

📋 Problem to solve

f(x,y)=(x^2-y^2,2xy)

locally invertible at

(1,0)

Exercise 4Surface areaHard

📋 Problem to solve

Area of the plane

z=x+y

over

[0,1]^2

Exercise 5LagrangeMedium

📋 Problem to solve

Min/max of

f=x^2+y^2

x^2+2y^2=1

Exercise 6Chain ruleHard

📋 Problem to solve

For

f(u,v)=(u^2,uv)

and

g(t)=(t,t^2)

, compute

(f\circ g)'(1)

📚

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