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Extending the Integral

Multivariable calculus is all about extending the concepts of calculus to functions of multiple variables. We've already seen how to differentiate functions of multiple variables, but what about integrating them?

Pages​

Overview​

We will start by discussing the line integral, a generalization of the definite integral. We connect the line integral to the concept of work, and show how it can be used to calculate the work done by a force along a curve. Throughout, we emphasize visual intuition and provide geometric interpretations of the line integral.

We stay in the realm of line integrals and introduce the gradient theorem, which is similar to the fundamental theorem of calculus, but for line integrals. Afterwards, we discuss conservative vector fields and how they relate to line integrals. We explore their properties, and rigorously prove that all path-independent vector fields can be expressed as the gradient of a scalar field.

After the line integral, we take a sneak peek at the concept of flux in two dimensions, and how it can be used to calculate the flow of a vector field across a curve.

Next, we delve into double integrals, which are the natural extension of the definite integral to functions of two variables. We discuss the different types of double integrals, and how they can be used to calculate the volume under a surface. However, we also show that double integrals, just like single-variable integrals, are not just a party trick for calculating areas and volumes; they have deep connections beyond that.

Finally, we end with the triple integral, which extends the concept of the definite integral to functions of three variables.