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4 Multivatiate calculus

1. Multivariate Calculus#

Multivariate calculus deals with functions of multiple variables. If f(x,y,z) is a function of multiple variables, the concepts of limits, continuity, partial derivatives, and multiple integrals extend to higher dimensions.

1.1 Partial Derivatives#

The partial derivative of a multivariable function is the derivative with respect to one variable, keeping the others constant.

For f(x,y)=x2y+3xy2:
- Partial derivative with respect to x:
[
\frac{\partial f}{\partial x} = 2xy + 3y^2
]
- Partial derivative with respect to y:
[
\frac{\partial f}{\partial y} = x^2 + 6xy
]

Example 1#

Find the partial derivatives of f(x,y,z)=x2y+yz3+z2x.

1.2 Gradient (∇f)#

The gradient of a scalar function f(x,y,z) is a vector pointing in the direction of the greatest rate of increase of the function:
[
\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)
]

Example 2#

Find the gradient of f(x,y,z)=3x2+4y2−z2.

1.3 Double and Triple Integrals#

Used to compute volumes and areas in higher dimensions:
- Double integral:
[
\iint_R f(x, y) \, dx \, dy
]
Example: Compute ∬Rx+ydxdy where R:0≤x≤1,0≤y≤1.

  • Triple integral:
    [
    \iiint_V f(x, y, z) \, dx \, dy \, dz
    ]

Example 3#

Evaluate ∭V(x+y+z)dxdydz, where V is a cube 0≤x,y,z≤1.

Practice Problems#

  1. Find the partial derivatives of f(x,y)=exy+sin⁡(x)cos⁡(y).
  2. Compute the gradient of f(x,y,z)=x2y−z3.
  3. Evaluate ∬Rx2ydxdy over R:0≤x≤1,0≤y≤2.

2. Vector Calculus#

Vector calculus deals with vector fields and operations like divergence, curl, and line integrals.

2.1 Vector Fields#

A vector field assigns a vector to each point in space. For example, F(x,y,z)=(yz,xz,xy) is a vector field.

2.2 Divergence#

Divergence of a vector field F=(F1,F2,F3):
[
\text{div} \, \mathbf{F} = \nabla \cdot \mathbf{F} = \frac{\partial F_1}{\partial x} + \frac{\partial F_2}{\partial y} + \frac{\partial F_3}{\partial z}
]

Example 4#

Find the divergence of F(x,y,z)=(x2y,y2z,z2x).

2.3 Curl#

Curl of a vector field F:
[
\text{curl} \, \mathbf{F} = \nabla \times \mathbf{F}
]
Example: Compute the curl of F(x,y,z)=(yz,xz,xy).

Practice Problems#

  1. Find the divergence of F(x,y,z)=(x+y,y+z,z+x).
  2. Compute the curl of F(x,y,z)=(x2,y2,z2).

3. Jacobian and Hessian#

3.1 Jacobian#

The Jacobian matrix contains the partial derivatives of a vector-valued function F(x,y,z)=(u(x,y),v(x,y)):
[
J = [∂u∂x∂u∂y ∂v∂x∂v∂y]
]

  • Jacobian determinant is used in coordinate transformations.

Example 5#

Compute the Jacobian of u=x2y,v=sin⁡(x)+y2.

3.2 Hessian#

The Hessian is a square matrix of second-order partial derivatives:
[
H = [∂2f∂x2∂2f∂x∂y ∂2f∂y∂x∂2f∂y2]
]

Example 6#

Find the Hessian of f(x,y)=x3+y3−3xy.

Practice Problems#

  1. Compute the Jacobian for u=x2+y2,v=xy.
  2. Find the Hessian of f(x,y)=ex+y+x2y.

4. Multivariate Taylor Series#

The Taylor expansion of a function f(x,y) around a point (a,b):
[
f(x, y) \approx f(a, b) + \frac{\partial f}{\partial x} (x-a) + \frac{\partial f}{\partial y} (y-b) + \frac{1}{2} \left( \frac{\partial^2 f}{\partial x^2} (x-a)^2 + 2\frac{\partial^2 f}{\partial x \partial y} (x-a)(y-b) + \frac{\partial^2 f}{\partial y^2} (y-b)^2 \right)
]

Example 7#

Find the Taylor expansion up to the second order of f(x,y)=x2+y2 around (0,0).

Practice Problems#

  1. Expand f(x,y)=exy around (0,0) up to the second order.
  2. Verify the Taylor series approximation for f(x,y)=sin⁡(xy) around (0,0).

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