Category Archives: Vector Calculus

Introduction to Double Integration

1. Consider . What is the volume of the region (1) The first thing we do: consider the rectangle and form a partition of into segments and into segments. This gives us a mesh of rectangles as specified by the … Continue reading

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Lagrange Multipliers

1. So, last time we considered finding extrema for some function , but what if we constrain our focus to some surface ? For example, the unit circle would have (1) How do we find extrema for on the unit … Continue reading

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Finding Extrema of Multivariable Functions

1. Remember for a curve , we have maxima and minima occur whenever (1) What’s the multivariable analog to this notion?

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Directional Derivative, Gradient

1. Suppose we have a scalar function of several variables (1) Let be some unit vector. How does change in the direction? We can consider this quantity as a function (2) What does this look like?

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Curves, Velocity, “Classical Kinematics”

1. Curves. We are interested in describing the motion of my car. Well, everyone is interested in the motion of my car. How can we describe it mathematically? First we approximate the car as a point. The point-like car moves … Continue reading

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Continuity for Functions of Several Variables, Partial Derivatives

1. We considered differentiating and integrating functions of a single-variable. How? We began with the notion of a limit, and then considered the derivative. If we have a, e.g., polynomial (1) we see (2) Again we stop and reflect: this … Continue reading

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Surfaces (…well, “Quadrics”)

1. Let where , , …, are all independent variables. Then the “Domain” of is the set of -tuples . Note that an ordered pair is (1) The set of corresponding values is the “Range” (or Codomain) of the function. … Continue reading

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