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Magazine Problem 61
Lateral Surface Area In Exercises \(61-68\), find the area of the lateral surface (see figure) over the curve \(C\) in the \(x y\) -plane and under the surface \(z=f(x, y)\), where Lateral surface area \(=\int_{C} f(x, y) d s .\) \(f(x, y)=h, \quad C\) : line from \((0,0)\) to \((3,4)\)
Problem 61
Find the divergence of the vector field \(\mathbf{F}\) at the given point. $$ \begin{array}{ll} \text { Vector Field } & \text { Point } \\ \hline \mathbf{F}(x, y, z)=x y z \mathbf{i}+y \mathbf{j}+z \mathbf{k} & (1,2,1) \end{array} $$
Problem 62
Find the divergence of the vector field \(\mathbf{F}\) at the given point. $$ \begin{array}{ll} \text { Vector Field } & \text { Point } \\ \mathbf{F}(x, y, z)=x^{2} z \mathbf{i}-2 x z \mathbf{j}+y z \mathbf{k} &(2,-1,3) \end{array} $$
Problem 62
Lateral Surface Area In Exercises \(61-68\), find the area of the lateral surface (see figure) over the curve \(C\) in the \(x y\) -plane and under the surface \(z=f(x, y)\), where Lateral surface area \(=\int_{C} f(x, y) d s .\) \(f(x, y)=y, \quad C:\) line from \((0,0)\) to \((4,4)\)
Problem 63
Find the divergence of the vector field \(\mathbf{F}\) at the given point. $$ \begin{array}{ll} \text { Vector Field } & \text { Point } \\ \mathbf{F}(x, y, z)=e^{x} \sin y \mathbf{i}-e^{x} \cos y \mathbf{j} &(0,0,3) \end{array} $$
Problem 63
Find the area of the lateral surface (see figure) over the curve \(C\) in the \(x y\) -plane and under the surface \(z=f(x, y)\), where Lateral surface area \(=\int_{C} f(x, y) d s .\) \(f(x, y)=x y, \quad\) C: \(x^{2}+y^{2}=1\) from \((1,0)\) to \((0,1)\)
Problem 64
Find the divergence of the vector field \(\mathbf{F}\) at the given point. $$ \begin{array}{ll} \text { Vector Field } & \text { Point } \\ \mathbf{F}(x, y, z)=\ln (x y z)(\mathbf{i}+\mathbf{j}+\mathbf{k}) &(3,2,1) \end{array} $$
Problem 64
Find the area of the lateral surface (see figure) over the curve \(C\) in the \(x y\) -plane and under the surface \(z=f(x, y)\), where Lateral surface area \(=\int_{C} f(x, y) d s .\) \(f(x, y)=x+y, \quad C: x^{2}+y^{2}=1\) from \((1,0)\) to \((0,1)\)
Problem 65
Find the area of the lateral surface (see figure) over the curve \(C\) in the \(x y\) -plane and under the surface \(z=f(x, y)\), where Lateral surface area \(=\int_{C} f(x, y) d s .\) \(f(x, y)=h, \quad C: y=1-x^{2}\) from \((1,0)\) to \((0,1)\)
Problem 65
Define a vector field in the plane and in space. Give some physical examples of vector fields.
