6/28/2023 0 Comments Vector calculus identitiesIf F is a sufficiently well-behaved vector field in spherical coordinates, show that ∇ × F is solenoidal. If F is a sufficiently well-behaved vector field in cylindrical coordinates, show that ∇ × F is solenoidal. Show that for sufficiently well-behaved functions f x, y, z, the gradient of f is curl-free. If f = x y 2 z 3, show that ∇ f is curl-free. If F = u x, y, z i + u x, y, z j + w x, y, z k, show that for sufficiently well-behaved u, v, w, the curl of F is solenoidal. Methods for finding scalar potentials for gradient fields, and vector potentials for solenoidal fields are postponed to Section 9.7, after the appropriate discussions of integration have taken place in Section 9.5 and Section 9.6. Finally, since the curl is a measure of rotation (or twist), the identity can be rephrased as "Gradients don't twist." A test for the existence of a scalar potential is the vanishing of the curl. The scalar whose gradient is the vector field is called a "scalar potential" for the vector field. Identity 2 shows that the curl of a gradient field is necessarily the zero vector so such fields are often called "curl-free". Finally, since the divergence is a measure of "spread," the identity can also be rephrased as "Curls don't spread." Thus, a test for whether a field has a vector potential is the vanishing of its divergence. A Vector field whose divergence vanishes is called solenoidal (or divergence-free), so Identity 1 can be rephrased as "Curl-fields are solenoidal." If a field is solenoidal, then it must be the curl of some other vector, and this vector is called a "vector potential" for the solenoidal field. Identity 1 shows that the divergence of any curl-field is necessarily zero. Special attention should be paid to Identities 1 and 2. identities 4 and 8 are difficult to remember, especially that both contain generalized directional-derivative operators such as F Identities 3, 5, and 7 suggest the "product-rule behavior" of "first times the derivative of the second plus the second times the derivative of the first." Identity 6 nearly follows this pattern, except for the minus sign. Table 9.4.1 Vector differential identities The eight left-hand sides of the identities listed in the table are the only combinations of the symbols that make sense, so that the given listing is exhaustive. The first two contain two del-operators on the left the remaining six, just one. Table 9.4.1 lists eight vector differential identities.
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