Podcast
Questions and Answers
What does the integrand $|{\bf r}_u\times{\bf r}_v|,du,dv$ represent?
What does the integrand $|{\bf r}_u\times{\bf r}_v|,du,dv$ represent?
The integrand represents the area of a tiny parallelogram, or a very small surface area.
How is the integral for surface area abbreviated?
How is the integral for surface area abbreviated?
The integral for surface area is abbreviated as $dS$.
What is the shortened version of the integral?
What is the shortened version of the integral?
The shortened version of the integral is $\dint{D} 1\cdot dS$.
What is the difference between $dS$ and $dA$?
What is the difference between $dS$ and $dA$?
Signup and view all the answers
How can integrals be computed over surfaces in space?
How can integrals be computed over surfaces in space?
Signup and view all the answers
What is a partial differential equation (PDE)?
What is a partial differential equation (PDE)?
Signup and view all the answers
Why are solutions to partial differential equations usually impossible to write down explicitly?
Why are solutions to partial differential equations usually impossible to write down explicitly?
Signup and view all the answers
What is the purpose of numerical approximation methods in relation to partial differential equations?
What is the purpose of numerical approximation methods in relation to partial differential equations?
Signup and view all the answers
What are some of the general qualitative features of solutions of partial differential equations that are studied in mathematical research?
What are some of the general qualitative features of solutions of partial differential equations that are studied in mathematical research?
Signup and view all the answers
What are some of the open questions in the field of partial differential equations?
What are some of the open questions in the field of partial differential equations?
Signup and view all the answers
Study Notes
Surface Integrals
- The surface integral for surface area is denoted as $$\int_a^b\int_c^d |{\bf r}_u\times{\bf r}_v|,du,dv$$
- The integrand $|{\bf r}_u\times{\bf r}_v|,du,dv$ represents the area of a tiny parallelogram, which is a very small surface area
- This integrand can be abbreviated as $dS$, making the shortened version of the integral $$\dint{D} 1\cdot dS$$
- The $dS$ hides more detail than $dA$, which is used for integrating functions over regions in the plane
Surface Integrals in Space
- We can compute integrals over surfaces in space using $$\dint{D} f(x,y,z), dS$$
- This requires a vector function ${\bf r}(u,v)=\langle x(u,v),y(u,v),z(u,v)\rangle$ for the surface
- The integral we compute is $$\int_a^b\int_c^d f(x(u,v),y(u,v),z(u,v)) |{\bf r}_u\times{\bf r}_v|,du,dv$$
Partial Differential Equations
- A partial differential equation (PDE) is an equation that computes a function between various partial derivatives of a multivariable function
- The function is often thought of as an "unknown" to be solved for
- It is usually impossible to write down explicit formulae for solutions of partial differential equations
- Numerical approximations of solutions can be obtained using computers
- Pure mathematical research on partial differential equations focuses on identifying general qualitative features of solutions, such as existence, uniqueness, regularity, and stability
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
Description
Test your understanding of surface integrals with this quiz! Learn how to calculate the area of a tiny parallelogram and abbreviate it as "dS" in the integral for surface area.
