physical significance of gradient divergence and curl pdf

Physical Significance Of Gradient Divergence And Curl Pdf

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Gradient of a scalar field the gradient of a scalar function fx1, x2, x3. Imagine a fluid, with the vector field representing the velocity of the fluid at each point in space.

In this final section we will establish some relationships between the gradient, divergence and curl, and we will also introduce a new quantity called the Laplacian. We will then show how to write these quantities in cylindrical and spherical coordinates. Note that this is a real-valued function, to which we will give a special name:. Notice that in Example 4.

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In this final section we will establish some relationships between the gradient, divergence and curl, and we will also introduce a new quantity called the Laplacian. We will then show how to write these quantities in cylindrical and spherical coordinates. Note that this is a real-valued function, to which we will give a special name:. Notice that in Example 4. Another way of stating Theorem 4.

Also, notice that in Example 4. There is another method for proving Theorem 4. The proof is not trivial, and physicists do not usually bother to prove it. But the result is true, and can also be applied to double and triple integrals. For instance, to prove Theorem 4. Often especially in physics it is convenient to use other coordinate systems when dealing with quantities such as the gradient, divergence, curl and Laplacian.

We will present the formulas for these in cylindrical and spherical coordinates. Recall from Section 1. We can now summarize the expressions for the gradient, divergence, curl and Laplacian in Cartesian, cylindrical and spherical coordinates in the following tables:.

The derivation of the above formulas for cylindrical and spherical coordinates is straightforward but extremely tedious. The basic idea is to take the Cartesian equivalent of the quantity in question and to substitute into that formula using the appropriate coordinate transformation.

As an example, we will derive the formula for the gradient in spherical coordinates. We can see from Figure 4. This comes down to solving a system of three equations in three unknowns. Lastly, we will solve for k. Again, this involves solving a system of three equations in three unknowns. Using a similar process of elimination as in Step 2, we get:.

But the algebra is straightforward and yields the desired result:. In Example 4. Verify that we get the same answers if we switch to spherical coordinates.

Michael Corral Schoolcraft College. Example 4. Theorem 4. Corollary 4.

What Is The Physical Meaning Of Divergence, Curl And Gradient Of A Vector Field_ - Quora

In this article learn about what is Gradient of a scalar field and its physical significance. We have also written an article on scalar and vector fields which is the topic you must learn before doing this topic. Let us consider a metal bar whose temperature varies from point to point in some complicated manner. So, the temperature will be a function of x, y, z in the Cartesian coordinate system. Hence temperature here is a scalar field represented by the function T x,y,z. Since temperature depends on distance it could increase in some directions and decrease in some directions. It could increase or decrease rapidly along with some directions in comparison to other directions.

Not logged in. More information may be available The course provides an elementary introduction to vector calculus and aims to familiarise the student with the basic ideas of the differential calculus the vector gradient, divergence and curl and the integral calculus line, surface and volume integrals and the theorems of Gauss and Stokes. The physical interpretation of the mathematical ideas will be stressed throughout via applications which centre on the derivation and manipulation of the common partial differential equations of engineering. The analytical solution of simple partial differential equations by the method of separation of variables will also be discussed.

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Both the divergence and curl are vector operators whose properties are revealed by viewing a vector field as the flow of a fluid or gas. Here we focus on the geometric properties of the divergence; you can read a similar discussion of the curl on another page. The divergence of a vector field is relatively easy to understand intuitively.

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