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- https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/03%3A_Multiple_IntegralsThe 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.
- 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.
- https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/02%3A_Functions_of_Several_Variables/2.04%3A_Directional_Derivatives_and_the_GradientFor 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.
- https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/02%3A_Functions_of_Several_Variables/2.03%3A_Tangent_Plane_to_a_SurfaceSince 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.
- 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.
- https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/01%3A_Vectors_in_Euclidean_Space/1.04%3A_Cross_ProductIn 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 R3. The definition may appear strange and lacking motivation, but we will see the geometric basis for it shortly.
- https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/03%3A_Multiple_Integrals/3.04%3A_Numerical_Approximation_of_Multiple_IntegralsFor 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.
- https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/02%3A_Functions_of_Several_VariablesIn 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.
- https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/03%3A_Multiple_Integrals/3.07%3A_Application-_Probability_and_Expectation_ValuesIn this section we will briefly discuss some applications of multiple integrals in the field of probability theory. In particular we will see ways in which multiple integrals can be used to calculate ...In this section we will briefly discuss some applications of multiple integrals in the field of probability theory. In particular we will see ways in which multiple integrals can be used to calculate probabilities and expected values.
- https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/01%3A_Vectors_in_Euclidean_Space/1.02%3A_Vector_AlgebraNow that we know what vectors are, we can start to perform some of the usual algebraic operations on them (e.g. addition, subtraction). Before doing that, we will introduce the notion of a scalar. T...Now that we know what vectors are, we can start to perform some of the usual algebraic operations on them (e.g. addition, subtraction). Before doing that, we will introduce the notion of a scalar. The term scalar was invented to convey the sense of something that could be represented by a point on a scale or graduated ruler. The word vector comes from Latin, where it means "carrier''. Examples of scalar quantities are mass, electric charge, and speed (not velocity).
- https://math.libretexts.org/Bookshelves/Calculus/Vector_Calculus_(Corral)/01%3A_Vectors_in_Euclidean_Space/1.06%3A_SurfacesA plane in Euclidean space is an example of a surface, which we will define informally as the solution set of the equation F(x,y,z)=0 in R3, for some real-valued function F. For example, a plane give...A plane in Euclidean space is an example of a surface, which we will define informally as the solution set of the equation F(x,y,z)=0 in R3, for some real-valued function F. For example, a plane given by ax+by+cz+d=0 is the solution set of F(x,y,z)=0 for the function F(x,y,z)=ax+by+cz+d. Surfaces are 2-dimensional. The plane is the simplest surface, since it is "flat''. In this section we will look at some surfaces that are more complex, the most important of which are spheres and the cylinders
