Lec 28: Divergence theorem | MIT 18.02 Multivariable Calculus, Fall 2007
TLDRThis script from an MIT OpenCourseWare lecture delves into the concept of flux in vector calculus, explaining how to compute it through surface integrals. It covers various methods to determine the normal vector and area element for different surfaces, including graphs of functions and parametric descriptions. The lecture also introduces the Divergence Theorem, offering an efficient way to calculate flux without surface integrals by relating it to the triple integral of the divergence of a vector field over a volume. The example of flux through a paraboloid and the physical interpretation of divergence as a measure of source expansion or sink contraction are highlighted.
Takeaways
- 📚 The script discusses the concept of flux through a surface S for a vector field F, defined by the integral of F dot n dS, where n is the unit normal to the surface and dS is the area element.
- 🌐 Flux calculations vary depending on the surface type, with specific formulas for different surfaces like spheres and horizontal planes, involving integration over angles or Cartesian coordinates.
- 📈 The script introduces a formula for the case where the surface S is the graph of a function z = f(x, y), leading to a double integral in terms of x and y, using the partial derivatives of the function.
- 📝 For a simple horizontal plane, the partial derivatives become zero, simplifying the area element to dx dy, which aligns with common sense.
- 📐 The method for finding the area of a small piece of surface and its normal vector involves using the cross-product of vectors corresponding to small changes in x and y directions on the surface.
- 📘 The cross-product not only provides the area element but also the normal direction, simplifying the process of finding n dS.
- 🔍 An example of flux calculation for a vector field through a paraboloid above a unit disk is given, illustrating the application of the formulas discussed.
- 🌀 The divergence theorem (also known as Gauss-Green theorem or Gauss theorem) is introduced as a way to compute flux through a closed surface without direct surface integrals.
- 🔄 The theorem relates the surface integral of a vector field to the triple integral of the divergence of the field over the enclosed volume, offering a more efficient calculation method.
- 🌍 The script explains that the divergence of a 3D vector field measures how much the flow is expanding or contracting at a given point, which is crucial for understanding the physical interpretation of the flux.
- 📚 The physical interpretation of the divergence theorem is highlighted, indicating that the total amount of a quantity flowing out of a region is equal to the total sources minus the sinks within that region.
Q & A
What is the definition of flux of a vector field through a surface S?
-The flux of a vector field F through a surface S is defined as the double integral over the surface of F dot n dS, where n is the unit normal to the surface and dS is the area element on the surface.
How do different surfaces affect the computation of flux?
-Different surfaces have different formulas for the normal vector and the area element, which affect how the flux is computed. For example, on a sphere, integration is typically done with respect to phi and theta for latitude and longitude angles, while on a horizontal plane, the area element simplifies to dx dy.
What is the significance of the formula for n dS when the surface is the graph of a function z = f(x, y)?
-The formula for n dS in the case of a surface graphed by z = f(x, y) allows us to set up and evaluate the integral in terms of x and y, simplifying the computation of flux by replacing z with f(x, y) wherever it appears in the vector field.
Why is the cross-product used to find the area of a small piece of surface and its normal vector?
-The cross-product is used because it provides both the area of the parallelogram formed by a small piece of the surface and its normal direction. It gives the magnitude of the area and the direction perpendicular to the surface, which is essential for calculating flux.
What is the physical interpretation of the divergence theorem?
-Physically, the divergence theorem represents the total amount of 'stuff' (like fluid or a vector field) that flows out of a closed region per unit time, which is equivalent to the total amount of sources minus the sinks within the region.
How does the divergence theorem simplify the calculation of flux through a closed surface?
-The divergence theorem allows the flux integral over a closed surface to be replaced by a triple integral of the divergence of the vector field over the enclosed volume, simplifying the computation by avoiding the need to compute surface integrals directly.
What is the formula for n dS in terms of the gradient of a function when the surface is given by z = f(x, y)?
-The formula for n dS when the surface is given by z = f(x, y) is n dS equals negative fx, negative fy, one, dxdy, where fx and fy are the partial derivatives of f with respect to x and y, respectively.
Can you provide an example of how to use the formula for n dS in a flux calculation?
-An example is given in the script where the flux of the vector field z times k (a vertical vector field) through the portion of the paraboloid z = x^2 y^2 above the unit disk is calculated. The calculation involves using the formula for n dS and integrating over the specified region.
What is the general formula for n dS when a surface is parameterized by two variables u and v?
-The general formula for n dS when a surface is parameterized by u and v is n dS equals the cross-product of (∂r/∂u) and (∂r/∂v) times du dv, where r is the position vector of the surface in terms of u and v.
How can the normal vector to a surface be determined if the surface is given by an implicit equation g(x, y, z) = 0?
-If a surface is given by an implicit equation g(x, y, z) = 0, the normal vector to the surface can be taken as the gradient of g, which is a vector with components ∂g/∂x, ∂g/∂y, and ∂g/∂z.
What is the relationship between the surface element dS and the area element dA in the xy-plane when dealing with a slanted plane?
-The relationship between the surface element dS and the area element dA in the xy-plane for a slanted plane is given by dA = dS * cos(alpha), where alpha is the angle the plane makes with the horizontal direction.
Outlines
📚 Introduction to Flux and Vector Field Surface Integrals
The paragraph introduces the concept of flux through a surface defined by the integral of a vector field F dot n dS, where n is the unit normal to the surface and dS is the area element. It discusses the different formulas for the normal vector and area element depending on the type of surface, such as spheres and planes. The lecturer also mentions a formula derived in the previous lecture, which will be explained in this session, and sets the stage for a detailed exploration of the topic.
📐 Derivation of the Formula for n dS on a Surface Defined by a Function
This paragraph delves into the derivation of the formula for the normal vector area element (n dS) when the surface S is the graph of a function z = f(x, y). It explains the process of finding the area of a small parallelogram on the surface using the cross-product of two vectors, U and V, which represent small changes in x and y, respectively. The paragraph concludes with the formula n dS = ±(-f_x dxdy - f_y dydx), highlighting the importance of the cross-product in determining both the area element and the normal direction.
🔍 Exploring the Flux of a Vector Field Through a Paraboloid
The lecturer presents an example of calculating the flux of a vertical vector field through a portion of a paraboloid above a unit disk. The vector field is given by z times the unit vector k. The discussion includes the orientation of the paraboloid and the expected sign of the flux. The lecturer uses the previously derived formula for n dS to set up the integral for the flux, emphasizing the importance of the normal vector's direction in determining the flux's sign.
📘 Setting Up the Flux Integral for a Paraboloid
This paragraph focuses on the setup for the flux integral of the vector field through the paraboloid, using the formula for n dS derived earlier. The lecturer substitutes the function z = x^2 y^2 into the formula to obtain the integrand and then transforms the integral into a double integral over the unit disk in the xy-plane. The use of polar coordinates simplifies the calculation, leading to the conclusion that the flux is pi/2.
🌐 Generalizing Flux Integrals for Parametric Surfaces
The discussion moves to more complex surfaces that cannot be expressed as z = f(x, y) but can be parameterized by two variables, u and v. The lecturer explains how to set up a flux integral on such a surface, introducing the concept of using the cross-product of partial derivatives with respect to u and v to find n dS in terms of du and dv. The paragraph emphasizes the generality of the method and its applicability to various surfaces.
📚 Understanding Surface Integrals with Given Normal Vectors
This paragraph addresses the scenario where the surface is defined by an equation, and a normal vector N is known, possibly from the gradient of a function defining the surface. The lecturer explains how to relate the surface element dS to the area in the xy-plane (delta A) by considering the cosine of the angle between the normal vector and the vertical direction. The final formula N dS = ±N/(N dot k) dx dy is presented, offering an alternative method for calculating surface integrals.
📘 Applying the Divergence Theorem to Flux Integrals
The lecturer introduces the divergence theorem, also known as Gauss-Green theorem, as a means to avoid direct computation of surface integrals for closed surfaces. The theorem states that the flux integral over a closed surface can be replaced by the triple integral of the divergence of the vector field over the enclosed volume. The paragraph provides an example using the vector field zk and a sphere, demonstrating how the theorem simplifies the calculation of flux.
🌌 The Physical Interpretation of Divergence in Flux Calculations
In the final paragraph, the lecturer offers a physical interpretation of the divergence theorem, relating it to the concept of sources and sinks in a fluid flow. The divergence measures the expansion or contraction of the flow, and the total flux out of a region is equated to the total amount of sources minus the sinks within that region. The paragraph concludes with a promise to prove the theorem and explore its applications in future lectures.
Mindmap
Keywords
💡Flux
💡Vector Field
💡Surface Integral
💡Unit Normal Vector
💡Area Element
💡Graph of a Function
💡Partial Derivatives
💡Cross-Product
💡Parametric Equation
💡Divergence Theorem
💡Divergence
Highlights
Introduction to the concept of flux and its computation through integrals for a vector field over a surface.
Explanation of the flux formula involving the dot product of the vector field, unit normal, and area element.
Different methods for calculating the normal vector and area element for various surfaces like spheres and planes.
Derivation of the formula for flux when the surface is the graph of a function, z = f(x, y).
Illustration of how to find the area of a small surface element using the cross-product of vectors.
Example of calculating flux for a simple horizontal plane where the partial derivatives become zero.
Detailed walkthrough of setting up the integral for flux through a paraboloid using the derived formula.
Discussion on the importance of the normal vector's direction in determining the sign of the flux.
Application of the flux formula to a vertical vector field through a specific portion of a paraboloid.
Use of polar coordinates to simplify the integration process in the context of a unit disk.
Clarification on how to obtain the negative partial derivatives for the formula when dealing with flux.
Introduction to the concept of parametrizing a surface for flux integrals when z cannot be expressed as a function of x and y.
Explanation of how to set up flux integrals using parametric equations of a surface.
Demonstration of the cross-product method to find the area element and normal vector for a parametrically defined surface.
Transition to using the divergence theorem as an alternative to surface integrals for flux calculations.
Physical interpretation of the divergence theorem relating to the concept of sources and sinks in a fluid flow.
Proof and applications of the divergence theorem to be discussed in subsequent lectures.
Transcripts
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