Line_integrals_for_scalar_functions_articles_
Rather than integrating along a straight line, such as the x-axis, we will now start thinking about meandering through space. This topic starts with arc length, which leads in nicely to the broader idea of line integration.
Surface_integral_preliminaries_videos_
Here, Sal covers some of the skills you need to be able to understand surface integrals.
Triple integrals are a way of integreating throughout a three-dimensional region in space.
Line_integrals_in_vector_fields_videos_
You've done some work with line integral with scalar functions and you know something about parameterizing position-vector valued functions. In that case, welcome! You are now ready to explore a core tool math and physics: the line integral for vector fields. Need to know the work done as a mass is moved through a gravitational field. No sweat with line integrals.
A single definite integral can be used to find the area under a curve. with double integrals, we can start thinking about the volume under a surface!
Finding line integrals to be a bit boring? Well, this tutorial will add new dimension to your life by explore what surface integrals are and how we can calculate them.
A single definite integral can be used to find the area under a curve. with double integrals, we can start thinking about the volume under a surface! More generally, double integrals are useful anytime you feel the need to add up infinitely many infinitely small quantities inside some two-dimensional region.
Line_integrals_in_vector_fields_articles_
After introducing line integrals in the context of scalar-valued functions, we see how to integrate along curves which wander through a vector field. This leads to a very beautiful extension of the fundamental theorem of calculus, known as the fundamental theorem of line integrals.
This is about as many integrals we can use before our brains explode. Now we can sum variable quantities in three-dimensions (what is the mass of a 3-D wacky object that has variable density)!
Learn how to compute surface integrals in a vector field, which involves constructing a unit normal vector to a surface. This lets you compute how much fluid flows through a given surface, which provides an intuition much more broadly applicable than just fluids.
Flux can be view as the rate at which "stuff" passes through a surface. Imagine a net placed in a river and imagine the water that is flowing directly across the net in a unit of time--this is flux (and it would depend on the orientation of the net, the shape of the net, and the speed and direction of the current). It is an important idea throughout physics and is key for understanding Stokes' theorem and the divergence theorem.
Just as line integrals give you the ability to add up points on a line, and double integrals give you the ability to add up points in a two-dimensional region, surface integrals are a mechanism for adding points on a curved surface in three-dimensional space.
Line_integrals_for_scalar_functions_videos_
With traditional integrals, our "path" was straight and linear (most of the time, we traversed the x-axis). Now we can explore taking integrals over any line or curve (called line integrals).
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