UQ-Advanced calculus

Juan Carlos Ponce Campuzano, Aug 29, 2016

Table of Contents

  1. Differentiation
    1. Partial derivatives and Tangent plane
    2. Directional derivatives and Gradient
  2. Integration
    1. Line integral for planar curves
    2. Line integrals of vector fields: Work & Circulation
    3. Line integrals of vector fields: Flux
    4. Double integral over a rectangle
    5. Line integrals: Arc length & Area of fence
    6. Surface integral of a vector field over a surface
  3. Vector fields
    1. Vector Fields in 2D
    2. Vector fields 3D
    3. Divergence and Curl calculator
    4. Vector fields 3D
  4. Velocity fields
    1. Basic examples of velocity fields
  5. Non-linear transformations
    1. Parabolic transformation
    2. Polar transformation

1.

Differentiation

  • 1. Partial derivatives and Tangent plane
  • 2. Directional derivatives and Gradient

1.1.

Partial derivatives and Tangent plane

This simulation shows the geometric interpretation of the partial derivatives of [i]f[/i]([i]x,y[/i]) at point A in [math]\mathbb{R}^2[/math]. It also shows the tangent plane at that point.[br][br]Things to try:                [list][*]Drag the point A in the xy-plane or type specific values on the boxes.              [/*][*]Select the object you want to show: Tangent plane, f[sub]x[/sub] or f[sub]y[/sub].           [/*][*]Use right click and drag the mouse to rotate the 3D view or click on View button.[/*][/list]
– Juan Carlos Ponce Campuzano

1.2.

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2.

Integration

  • 1. Line integral for planar curves
  • 2. Line integrals of vector fields: Work & Circulation
  • 3. Line integrals of vector fields: Flux
  • 4. Double integral over a rectangle
  • 5. Line integrals: Arc length & Area of fence
  • 6. Surface integral of a vector field over a surface

2.1.

Line integral for planar curves

– Juan Carlos Ponce Campuzano

2.2.

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2.3.

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2.4.

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2.5.

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2.6.

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3.

Vector fields

  • 1. Vector Fields in 2D
  • 2. Vector fields 3D
  • 3. Divergence and Curl calculator
  • 4. Vector fields 3D

3.1.

Vector Fields in 2D

Instructions:
1. Change the components of the vector field by typing, for example:[br][br] [math]F_1=[/math] x^2sin(y) , [math]F_2=[/math] sqrt(y^2+x)exp(x/y)[br][br]2. Change the [b]Scale[/b] or [b]Vectors density[/b] to provide a better visualisation of the vector field. [br]3.[b] Zoom In[/b] or [b]Out[/b] (or drag the plane) to change the domain.
– Juan Carlos Ponce Campuzano

3.2.

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3.3.

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3.4.

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4.

Velocity fields

  • 1. Basic examples of velocity fields

4.1.

Basic examples of velocity fields

– Juan Carlos Ponce Campuzano

5.

Non-linear transformations

  • 1. Parabolic transformation
  • 2. Polar transformation

5.1.

Parabolic transformation

Instructions:
A transformation is defined by the equations [math]x=u^2-v^2[/math] and [math]y=2uv[/math]. [br][br]1. Drag the [color=#3d85c6]BLUE[/color] point defined on the boundary of the square (left-side). Observe the effect of the transformation on the right side.[br][br]2. Drag the [color=#ff0000]RED[/color] point to change the position of the square. Observe the effect of the transformation on the right side.
– Juan Carlos Ponce Campuzano

5.2.

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