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3: Multiple Integrals
https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/03%3A_Multiple_Integrals
The multiple integral is a generalization of the definite integral with one variable to functions of more than one real variable. For definite multiple integrals, each variable can have different limi...The multiple integral is a generalization of the definite integral with one variable to functions of more than one real variable. For definite multiple integrals, each variable can have different limits of integration.
3.E: Multiple Integrals (Exercises)
https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/03%3A_Multiple_Integrals/3.E%3A_Multiple_Integrals_(Exercises)
Problems and select solutions to the chapter.
4.E: Line and Surface Integrals (Exercises)
https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/04%3A_Line_and_Surface_Integrals/4.E%3A_Line_and_Surface_Integrals_(Exercises)
Problems and select solutions to the chapter.
2.4: Directional Derivatives and the Gradient
https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/02%3A_Functions_of_Several_Variables/2.04%3A_Directional_Derivatives_and_the_Gradient
For a function z=f(x,y), we learned that the partial derivatives ∂f/∂x and∂f/∂y represent the (instantaneous) rate of change of f in the positive x and y directions, respectively. What about other dir...For a function z=f(x,y), we learned that the partial derivatives ∂f/∂x and∂f/∂y represent the (instantaneous) rate of change of f in the positive x and y directions, respectively. What about other directions? It turns out that we can find the rate of change in any direction using a more general type of derivative called a directional derivative.
2.3: Tangent Plane to a Surface
https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/02%3A_Functions_of_Several_Variables/2.03%3A_Tangent_Plane_to_a_Surface
Since the derivative dy/dx of a function y=f(x) is used to find the tangent line to the graph of f (which is a curve in R2), you might expect that partial derivatives can be used to define a tangent p...Since the derivative dy/dx of a function y=f(x) is used to find the tangent line to the graph of f (which is a curve in R2), you might expect that partial derivatives can be used to define a tangent plane to the graph of a surface z=f(x,y). This indeed turns out to be the case. First, we need a definition of a tangent plane. The intuitive idea is that a tangent plane “just touches” a surface at a point. The formal definition mimics the intuitive notion of a tangent line to a curve.
1.4: Cross Product
https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/01%3A_Vectors_in_Euclidean_Space/1.04%3A_Cross_Product
In Section 1.3 we defined the dot product, which gave a way of multiplying two vectors. The resulting product, however, was a scalar, not a vector. In this section we will define a product of two vect...In Section 1.3 we defined the dot product, which gave a way of multiplying two vectors. The resulting product, however, was a scalar, not a vector. In this section we will define a product of two vectors that does result in another vector. This product, called the cross product, is only defined for vectors in $\mathbb{R}^{3}$ . The definition may appear strange and lacking motivation, but we will see the geometric basis for it shortly.
3.4: Numerical Approximation of Multiple Integrals
https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/03%3A_Multiple_Integrals/3.04%3A_Numerical_Approximation_of_Multiple_Integrals
For complicated functions, it may not be possible to evaluate one of the iterated integrals in a simple closed form. Luckily there are numerical methods for approximating the value of a multiple integ...For complicated functions, it may not be possible to evaluate one of the iterated integrals in a simple closed form. Luckily there are numerical methods for approximating the value of a multiple integral. The method we will discuss is called the Monte Carlo method. The idea behind it is based on the concept of the average value of a function, which you learned in single-variable calculus.
2: Functions of Several Variables
https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/02%3A_Functions_of_Several_Variables
In the last chapter we considered functions taking a real number to a vector, which may also be viewed as functions, that is, for each input value we get a position in space. Now we turn to functions ...In the last chapter we considered functions taking a real number to a vector, which may also be viewed as functions, that is, for each input value we get a position in space. Now we turn to functions of several variables, meaning several input variables, functions.
4.4: Surface Integrals and the Divergence Theorem
https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/04%3A_Line_and_Surface_Integrals/4.04%3A_Surface_Integrals_and_the_Divergence_Theorem
We will now learn how to perform integration over a surface in $\mathbb{R}^3$ , such as a sphere or a paraboloid. Recall from Section 1.8 how we identified points $(x, y, z)$ on a curve $C$ in \...We will now learn how to perform integration over a surface in $\mathbb{R}^3$ , such as a sphere or a paraboloid. Recall from Section 1.8 how we identified points $(x, y, z)$ on a curve $C$ in $\mathbb{R}^3$ , parametrized by $x = x(t), y = y(t), z = z(t), a ≤ t ≤ b$ , with the terminal points of the position vector.
2.1: Functions of Two or Three Variables
https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/02%3A_Functions_of_Several_Variables/2.01%3A_Functions_of_Two_or_Three_Variables
We will now examine real-valued functions of a point (or vector) in $\mathbb{R}^2$ or $\mathbb{R}^3$ . For the most part these functions will be defined on sets of points in $\mathbb{R}^2$ ,...We will now examine real-valued functions of a point (or vector) in $\mathbb{R}^2$ or $\mathbb{R}^3$ . For the most part these functions will be defined on sets of points in $\mathbb{R}^2$ , but there will be times when we will use points in $\mathbb{R}^3$ , and there will also be times when it will be convenient to think of the points as vectors (or terminal points of vectors).
3.5: Change of Variables in Multiple Integrals
https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/03%3A_Multiple_Integrals/3.05%3A_Change_of_Variables_in_Multiple_Integrals
Given the difficulty of evaluating multiple integrals, the reader may be wondering if it is possible to simplify those integrals using a suitable substitution for the variables. The answer is yes, tho...Given the difficulty of evaluating multiple integrals, the reader may be wondering if it is possible to simplify those integrals using a suitable substitution for the variables. The answer is yes, though it is a bit more complicated than the substitution method which you learned in single-variable calculus.

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