Vector Calculus Juan

Juan Carlos Ponce Campuzano, Ronan Downes, Mar 16, 2020

Geometrical approach to Vector Calculus

Table of Contents

  1. Geometry of Euclidean Space and real-valued functions
    1. Addition of vectors
    2. Scalar multiplication
    3. Vector product
    4. Plotting 3D surfaces
    5. Parametric surfaces
    6. Klein bottle
    7. Distance: Point to Plane
  2. Differentiation
    1. Partial derivatives and Tangent plane
    2. Directional derivatives and Gradient
  3. Vector-Valued Functions
    1. Vector fields in 2D
    2. Vector field 3D
    3. Dynamic Frenet-Serret frame
    4. Vector Fields
    5. Divergence and Curl calculator
  4. Double integrals
    1. Double integral over a rectangle
  5. Integrals over paths and surfaces
    1. Path integral for planar curves
    2. Area of fence Example 1
    3. Line integral: Work
    4. Line integrals: Arc length & Area of fence
    5. Surface integral of a vector field over a surface
    6. Line integrals of vector fields: Work & Circulation
    7. Line integrals of vector fields: Flux
    8. Line integral for planar curves

1.

Geometry of Euclidean Space and real-valued functions

  • 1. Addition of vectors
  • 2. Scalar multiplication
  • 3. Vector product
  • 4. Plotting 3D surfaces
  • 5. Parametric surfaces
  • 6. Klein bottle
  • 7. Distance: Point to Plane

1.1.

Addition of vectors

Addition of two vectors.
Drag the vectors around. You can change the magnitude and direction of the vectors too. To show the parallelogram rule, place the vectors at the origin.
– Juan Carlos Ponce Campuzano

1.2.

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1.3.

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1.4.

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1.5.

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1.6.

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1.7.

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2.

Differentiation

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

2.1.

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

2.2.

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3.

Vector-Valued Functions

1. Vectors fields. 2. Frenet-Serret frame.

  • 1. Vector fields in 2D
  • 2. Vector field 3D
  • 3. Dynamic Frenet-Serret frame
  • 4. Vector Fields
  • 5. Divergence and Curl calculator

3.1.

Vector fields in 2D

– Juan Carlos Ponce Campuzano

3.2.

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3.3.

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3.4.

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3.5.

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4.

Double integrals

  • 1. Double integral over a rectangle

4.1.

Double integral over a rectangle

– Juan Carlos Ponce Campuzano

5.

Integrals over paths and surfaces

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

5.1.

Path integral for planar curves

This worksheet shows a geometrical representation of the path integral for planar curves. [br]If [math]c:[a,b]\rightarrow \mathbb R^2[/math] is of class [math]C^1[/math] and the composite function [math]t\rightarrow f(x(t),y(t))[/math] is continuous, then we define[br][math]\quad \quad\quad\int_c fds=\int_a^bf(x(t),y(t))||c'(t)||dt[/math][br]When [math]f(x,y)\geq0[/math], this integral has a geometric interpretation as the [b]area of a fence[/b].
Path integral for planar curves
Move the points in the plane to change the shape of the curve. You can change the surface as well.
– Juan Carlos Ponce Campuzano

5.2.

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5.3.

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5.4.

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5.5.

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5.6.

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5.7.

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5.8.

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