Multivariable Calculus

Kristen Beck, BurhanMasihBelajar, Sep 25, 2018

Table of Contents

  1. Parametric Equations
    1. Parametric Equations
    2. Different Parameterizations
    3. Parameterizing an Ellipse
    4. Curves in 2-Space
    5. Curves in 3-Space
    6. Tangent Line to a Parametric Curve
    7. Regions Bounded by Polar Curves
  2. Conic Sections
    1. The Foci of an Ellipse
    2. The Foci of a Hyperbola
    3. The Focus and Directrix of a Parabola
  3. Vectors
    1. Sum/Difference of Vectors
    2. Parametric Representation of a Line
    3. Vector Representation of a Line
    4. Intersecting Lines
    5. Geometric Interpretation of the Dot Product
    6. Orthogonal Projection
    7. Geometric Interpretation of the Cross Product
  4. Curves and Surfaces
    1. Level Curves
    2. Cylindrical Coordinates
    3. Spherical Coordinates
    4. Curve Along a Surface
  5. Partial Differentiation
    1. Partial Derivatives and Slope
    2. The Chain Rule
    3. Tangent Plane to a Surface
  6. Gradients and Optimization
    1. Gradient vs. Level Curves
    2. Constrained Optimization
  7. Multiple Integration
    1. Small Cylindrical Volumes
    2. Small Spherical Volumes
    3. Iterated Double Integrals
    4. Line Integral of a Scalar Function
    5. Simple Cylindrical Volumes
    6. Simple Spherical Volumes
  8. Vector Analysis
    1. Vector Field in the Plane
    2. Line Integral of a Vector Field in 2-Space
    3. Line Integral of a Vector Field in 3-Space
    4. Circulation in 2-Space

1.

Parametric Equations

  • 1. Parametric Equations
  • 2. Different Parameterizations
  • 3. Parameterizing an Ellipse
  • 4. Curves in 2-Space
  • 5. Curves in 3-Space
  • 6. Tangent Line to a Parametric Curve
  • 7. Regions Bounded by Polar Curves

1.1.

Parametric Equations

– Kristen Beck

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.

Conic Sections

  • 1. The Foci of an Ellipse
  • 2. The Foci of a Hyperbola
  • 3. The Focus and Directrix of a Parabola

2.1.

The Foci of an Ellipse

This worksheet illustrates the relationship between an ellipse and its foci. Move the yellow point along the ellipse.
What are the red points called?
What is the relationship between each of the green (resp. blue) line segments?
What happens to the [i]sum [/i]of the lengths of the green and blue line segments as the yellow point moves along the ellipse?
– Kristen Beck

2.2.

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2.3.

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3.

Vectors

  • 1. Sum/Difference of Vectors
  • 2. Parametric Representation of a Line
  • 3. Vector Representation of a Line
  • 4. Intersecting Lines
  • 5. Geometric Interpretation of the Dot Product
  • 6. Orthogonal Projection
  • 7. Geometric Interpretation of the Cross Product

3.1.

Sum/Difference of Vectors

This worksheet gives a geometric interpretation of the sum and difference of two vectors. (The yellow points may be moved.)
How do the dashed vectors compare to the solid vectors?
What is the significance of the green vector?
What is the significance of the black vector?
– Kristen Beck

3.2.

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3.3.

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3.4.

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3.5.

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3.6.

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3.7.

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4.

Curves and Surfaces

  • 1. Level Curves
  • 2. Cylindrical Coordinates
  • 3. Spherical Coordinates
  • 4. Curve Along a Surface

4.1.

Level Curves

This worksheet illustrates the level curves of a function of two variables. You may enter any function which is a polynomial in both [math]x[/math] and [math]y[/math].
– Kristen Beck

4.2.

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4.3.

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4.4.

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5.

Partial Differentiation

  • 1. Partial Derivatives and Slope
  • 2. The Chain Rule
  • 3. Tangent Plane to a Surface

5.1.

Partial Derivatives and Slope

This worksheet illustrates the connection between partial derivatives and slope. Move the yellow point (in the xy-plane) and compare the value of the partial derivative to the slope of the black line.
– Kristen Beck

5.2.

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5.3.

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6.

Gradients and Optimization

  • 1. Gradient vs. Level Curves
  • 2. Constrained Optimization

6.1.

Gradient vs. Level Curves

Move the yellow point to see how the gradient of the function (whose graph is the red surface) changes. Note that (1) the gradient is always perpendicular to the level curves, (2) though the gradient points in the direction of greatest change, this direction is not always directly toward the "summit" (i.e. the local maximum), and (3) the magnitude (length) of the gradient is equal to the slope of the tangent line to the surface in the direction of the gradient (this is the solid black line on the right side).
– Kristen Beck

6.2.

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7.

Multiple Integration

  • 1. Small Cylindrical Volumes
  • 2. Small Spherical Volumes
  • 3. Iterated Double Integrals
  • 4. Line Integral of a Scalar Function
  • 5. Simple Cylindrical Volumes
  • 6. Simple Spherical Volumes

7.1.

Small Cylindrical Volumes

Adjust the values of [math]r[/math], [math]\theta[/math], [math]z[/math], [math]\Delta r[/math], [math]\Delta\theta[/math], and [math]\Delta z[/math] to see how the small cylindrical volume changes. Note that this volume has approximate dimensions [math]\Delta r \times (r\Delta\theta) \times \Delta z= r\Delta r\Delta\theta\Delta z[/math].
Small Cylindrical Volumes
– Kristen Beck

7.2.

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7.3.

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7.4.

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7.5.

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7.6.

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8.

Vector Analysis

  • 1. Vector Field in the Plane
  • 2. Line Integral of a Vector Field in 2-Space
  • 3. Line Integral of a Vector Field in 3-Space
  • 4. Circulation in 2-Space

8.1.

Vector Field in the Plane

This worksheet illustrates a (scaled) vector field in the [math]xy[/math]-plane. Specify the vector field by defining the component functions [math]P(x,y)[/math] and [math]Q(x,y)[/math] and scaling constant [math]k[/math]. Move the position of the black point to observe the "value" of the vector field at that point.
Vector Field in the Plane
– Kristen Beck

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8.3.

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8.4.

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