How to of the Day

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• How to Treat a Stye
• How to Accept Honest Criticism from Your Partner
• How to Calculate Divergence and Curl

How to Treat a Stye Posted: 05 Sep 2016 05:00 PM PDT A stye, or hordeolum, is a small, tender lump which develops on the eyelid, causing it to swell. Styes can be painful and unsightly, but they are usually not serious and can be effectively treated at home. If at-home methods such as ointments or hot compresses are not helpful, medical help may be needed. If you have any vision changes along with a stye, or you have had any recent trauma to your face or eyes, immediately see your doctor or go to the nearest emergency room. EditSteps EditIdentifying Styes Look for a tender red lump on your eyelid. Styes are similar to pimples that form on the eyelid. Styes occur when bacteria (usually staphylococcus or "staph" bacteria) that live on the skin's surface cause infections in or near the eye. You can usually identify a stye from its appearance -- a tender red lump on the eyelid which may develop a small pus-filled center.[1] The stye develops inside the lid and protrudes either outward or inward. External styes involve Zeis or Moll glands and protrude out. Internal styes involve the meibomian gland and tend to protrude inward. An internal stye is just an infected chalazion.[2] Learn to recognize styes. Styes can be confused with chalazions, which also appear as a bump near the eye. The treatment of a stye and a chalazion is the same, so it's not a problem if you can't tell them apart. A stye or hordeolum is caused by bacterial infection. These red lumps are often filled with pus, like a pimple. Styes cause pain and swelling of the eyelid.[3] A chalazion is a cyst or fluid-filled sac that develops in the small oil glands (meibomian glands) near your eyelids. They more often appear at the inner side of the eyelid. Unlike styes, they are painless.[4] Blepharitis is an inflammation of the eyelid. Blepharitis have many causes, including infection, allergies, and untreated chalazions, Blepharitis makes the edges of your eyelids become red and swollen. Your eyelids may be itchy, crusty, or difficult to open.[5] Look for swelling. Your eyelid will usually swell up as the stye develops. In some cases the swelling may cause your eyelid to close completely. The swelling will usually last for about three days, after which it will sometimes burst and drain on its own. Occasionally, the inflammation will spread to areas of the face surrounding the eye, in which case you should see a doctor.[6] Be aware of other symptoms. The swollen eyelid can irritate the eye, causing pain and watering. Your eyes may also be more sensitive to light. The swelling may cause discomfort when you blink.[7] EditTreating Styes at Home Apply a warm compress. The recommended home treatment for a stye is to apply a warm compress to the eye for 10 to 15 minutes, 2 to 3 times a day.[8] This will increase circulation to the area and speed up the rupture and drainage of the stye. It is important to start warm compress treatment as soon as you identify the stye. Untreated, styes can last for several weeks, but treatment can cut this to just a few days.[9] Always use a warm compress on a closed eyelid. Make sure the compress is not too hot, or you might cause irritation or burns. Test the compress on a clean area of skin such as your arm for a few minutes. If it is too hot, let it cool for a few minutes and test it again before applying it to your face. To make a warm compress, wet a clean towel with hot (not boiling) water and wring out the excess. Lie down and apply the towel to your closed eyelid for 10 to 15 minutes. Reheat the towel if necessary. Alternatively, you can wrap a wet towel around a hot water bottle. Lie down and lay the bottle on your face with your eyes closed. This method keeps the towel hot longer. You can also use a microwaveable heating pack from the back pain aisle of the drug store. Heat it up so that it is warm to the touch but can be comfortably held. If you can't hold it in your hand, it is too hot for your face. These stay warm for about 30 minutes. Some people prefer to use warm, moist teabags on their eyelids. These are small and hold the heat well, but are no more effective than any damp cloth.[10] Use an over-the-counter treatment. There are many creams, ointments and even medicated pads available over-the-counter that will help to clear up the eye infection causing the stye. Look for treatments that contain the ingredient polymyxin B sulfate, which is an effective antibiotic in treating eye infections.[11] Make sure it says that it's safe for use in the eye. Ophthalmologists do not like OTC antibiotics because they are not safe if they get in the eye, which these ointments always do. Be wary of any product that claims to remove styes immediately or overnight. They are usually expensive and are unlikely to work as quickly as they claim. Stye treatment usually takes between 2-4 days.[12] Avoid squeezing the stye. Although it may be tempting to squeeze the stye or pop it with a needle in an attempt to drain it, never do so. Rupturing the stye can make the infection worse by introducing bacteria from your skin into the stye.[13] Apply a regular hot compress and wait for the stye to rupture and drain on its own. Attempting to pop or rupture the stye can actually spread the infection from the stye to other parts of your eye.[14] Stop wearing eye makeup. While you are suffering from a stye it is best to steer clear of eye make up such as mascara, eye liner and eye shadow. This will help to avoid irritating your eye and decrease your chances of reinfection. Since the infection is often present before you are aware of the symptoms, it is best to throw away any eye makeup applicators that you were using when the stye developed. This will help keep you from reinfecting your eye.[15] Avoid wearing contact lenses. It is a good idea to hold off on wearing contacts while you're suffering from a stye. This is because contacts could irritate your eye further, causing unnecessary pain. Switch to wearing normal glasses until the stye has cleared. Wearing contacts that you wore during the period of infection can reinfect your eye. Once your stye has cleared up, wear a new pair of contacts to keep your eyes healthy.[16] Keep your hands clean. Wash your hands with antibacterial soap and hot water before and after touching your eyes. If you touch your eyes with unwashed hands you risk transferring bacteria to the unaffected eye. You might even worsen the initial infection. Take over-the counter medication for pain relief. Over-the-counter non-steroidal pain medication, such as NSAIDs like ibuprofen or naproxen, can help to ease inflammation and pain caused by the stye. Always follow package instructions. If you are sensitive or allergic to ibuprofen or naproxen, acetaminophen can help relieve pain caused by the stye. It is not as effective at fighting inflammation as NSAIDs.[17] Always check with your physician or pharmacist to prevent possible interactions with everyday prescriptions. Give your eyes a break. It is important that your eyes get adequate rest while you are treating a stye. As you sleep, your immune system releases proteins called cytokines, which help fight infection and inflammation.[18] Get plenty of sleep at night and avoid doing too much reading or other visually-detailed work. Reducing strain on the eyes can help relieve the pain styes cause while you're awake, such as the pain caused by blinking. Rest your eyes frequently during work to give them short breaks. Close your eyes for a few minutes and try to relax. Treat a stye as you would if you were sick with other infections: get plenty of rest, drink lots of clear fluids, and take appropriate pain medications. Do not "overdose" on vitamin supplements, as this will not increase your immune response. However, taking a daily multivitamin is a good idea for maintaining a healthy immune system.[19] EditSeeking Medical Treatment See a doctor if the stye does not improve within two days. While full treatment can take up to four days, you should see a doctor if your stye has not begun to improve after 48 hours of treatment.[20] You should see a doctor immediately if you experience changes in your vision. Also seek immediate medical help if you experience any change in the amount or type of discharge from your eye.[21] Get a prescription for antibiotics. The doctor may prescribe a topical antibiotic, such as an ointment or eye drops, to clear up the infection. The doctor may also opt to prescribe oral antibiotics, especially when the styes are recurring.[22] Have a doctor lance the stye. If the stye has not cleared up after several days and shows no signs of rupturing on its own, a doctor may lance the stye. Lancing allows the stye to drain. This will usually be done with a local anesthetic in the doctor's office.[23] If the stye is affecting a baby or small child, the lancing may be done under general anesthetic.[24] Always let a trained medical professional lance styes. The eye is a very delicate area. Attempts to lance or puncture a stye on your own could cause infection or permanent damage to your eye.[25] EditPreventing Styes Minimize contact with your eyes. Bacteria, especially staphylococcus or "staph" bacteria, are often the culprit behind styes.[26] Rubbing your eyes can cause irritation and can transfer bacteria from your hands to your eyes. If you absolutely have to touch your eyes, make sure to wash your hands thoroughly with an anti-bacterial soap first. Be careful with eye make up. Old or expired makeup, especially liquid-based makeup, can be a breeding ground for bacteria. Avoid sharing eye makeup or tools with others. Always remove eye makeup before going to sleep at night. Replace mascara and liquid eyeliners every 2-3 months.[27] Foundation can also support the growth of bacteria, including staphylococcus, one of the culprits that causes styes. Replace liquid foundation every 6-12 months.[28] Clean makeup tools regularly. Clean your makeup tools and applicators after every few uses to keep germs away. Tools like eyelash curlers and eyeshadow brushes should be cleaned and disinfected regularly.[29] Wash metal tools such as eyelash curlers and tweezers with antibacterial soap and hot water. Wipe them down with a cotton ball soaked in rubbing alcohol to disinfect. Rinse brushes with baby shampoo and warm water. Disinfect them by soaking them in a solution of 2 parts water and 1 part white vinegar. Do not pump your mascara wand in and out of the bottle. Rinse your mascara wand in eye makeup cleaner and allow it to dry before putting it back in the bottle.[30] Protect your eyes from dust and other debris. Dirt and small objects can get trapped beneath the eyelid and cause infection. Protect your eyes whenever necessary by wearing safety goggles or sunglasses.[31] For example, you should protect your eyes when you are dusting. Wear eye protection when doing outside work, such as raking or mowing the lawn, or when doing shop work at home. If you are in a job that requires safety glasses, wear them! Eye injuries are a leading cause of workplace injury.[32] Scrub your eyelids. Some people have higher levels of oil build-up on their eyelids. This makes them more prone to styes. Regular lid scrubs can help prevent styes by removing excess bacteria and dead skin cells. They can also help reduce reinfection if you've already had a stye.[33] Make your own scrub solution with baby shampoo and warm water. Put a few drops of baby shampoo into a small cup and add warm water. Stir. Use a cotton swab or clean washcloth to rub the solution across your eyelids. Gently rub back and forth, as it is the rubbing motion that removes buildup. Commercial lid scrubs are available at most drug stores. Find one that will not irritate your eyes.[34] Consider whether you might be more susceptible to developing a stye. Although styes are very common and they can affect anyone, there are certain conditions which might make you more susceptible to getting one:[35] People with chronic conditions such as diabetes, seborrhea and blepharitis are more likely to develop styes, along with people with chronic debilitating diseases. People who have acne, rosacea, or seborrheic dermatitis are more likely to experience meibomian gland dysfunction, which can cause chalazions to develop.[36] People with high levels of blood lipids are at increased risk. Their oil glands are more likely to become blocked, which can cause styes.[37] People with high levels of stress may be more likely to develop styes.[38] EditTips Visit your eye doctor regularly. This can help you prevent styes and other infections. Regular eye exams can also help detect serious eye diseases such as glaucoma. Children should have eye exams every year. Adults should have eye exams at least every two years.[39] Wash your hands with antibacterial soap and hot water before touching your eyes. Throw away any contacts or makeup you used directly before developing the stye. This will help you to avoid reinfection. When the stye bursts, it may create secondary infections on the same or other eye. Treat them the same way. Try to keep the item you use to apply heat as clean as possible. It is best if you use it and then wash it thoroughly between heat applications if possible. EditWarnings Do not attempt to pinch, squeeze, puncture or otherwise force the stye to drain. This could spread the infection. It might also result in permanent damage to your eye. See a doctor immediately if you experience vision changes, double vision, "halos" (bright fields around objects), or a change in the type or amount of discharge. These could be signs of a more serious issue. EditSources and Citations Cite error: <ref> tags exist, but no <references/> tag was found

How to Accept Honest Criticism from Your Partner Posted: 05 Sep 2016 09:00 AM PDT Criticism is hard to hear at any time. But it's especially tough when it comes from your partner, someone that you love and respect. To accept honest criticism from your partner, remember that the aim is not to tear you down but to build up your relationship together. To do this you'll need to lower your defenses, seriously listen to the criticism, and try to empathize. EditSteps EditLowering Your Defenses Remind yourself it's not a zero-sum game. No one likes to be criticized. It can be very hard to hear how you've failed to meet your partner's expectations – you might feel unfairly blamed, misunderstood, or attacked. However, think about the reason you and your partner are communicating in the first place.[1] Keep in mind that your relationship shouldn't be a power-struggle. The fact that your partner has an honest criticism doesn't mean that you are "losing." Tell yourself, in the same way, that criticism isn't a zero-sum game. The point is to establish some give and take and hopefully improve your relationship. Don't interrupt. You'll never be able to accept honest criticism from a partner unless you start to lower your defenses, listen, and empathize. Give your partner the floor, for one thing. Don't interrupt to deny, refute, or turn back the criticism. This is a huge sign that you're being defensive.[2] Try this trick: count to ten every time that you are tempted to interrupt. There's a fair chance that once you're done the moment will have passed and your point won't matter anymore. Count higher if you still feel the urge, to twenty or thirty. Stop yourself deliberately and apologize if you do interrupt. Cut yourself off, express regret for your rudeness, and let your partner resume. Disarm other defensive tactics. People have a whole arsenal of defensive tactics that they use to ignore or deflect criticism. Learn to recognize what these are, how you use them, and how to disarm them. Once you've done so, you'll be better able to listen and create an open dialogue for honest criticism.[3] Do you placate or outright deny, saying "Sure honey, whatever you say" or "No, I don't do anything of the sort"? Or do you ignore the criticism or stonewall: "Well, that's just how I am. Deal with it." None of these tactics allow for an exchange of honest criticism. Some defensive tactics are less obvious and rely on manipulation. They can include invalidation – "Why are you making such a big deal out of this? – as well as guilting, for instance, "Why are you being so cruel? What about how I feel?" Other tactics shift responsibility to the criticizer: "Maybe I wouldn't have to be this way if you were nicer." Notice any other tactics you might use, like excuses or neutralizing the criticism – "I didn't mean to upset you. You've taken it the wrong way." Pay attention to your non-verbal signals. One big way that people communicate how they feel is through non-verbal signals like body language. Note how you're carrying yourself, not just what you're saying to your partner – this means body language, tone of voice, facial expression, eye contact, and how far away you're sitting.[4] Maintain eye contact with your partner. Looking away will make you seem standoffish, disinterested, or ashamed. Try not to fold your arms, either, or to face away from your partner. This gives the impression that you're defensive and closed off. Be aware of your of your facial expression and try to remain neutral. A raised eyebrow or pursed lips can signal judgement or disagreement to your partner. Use a normal, steady, conversational tone of voice. Raising your voice signals that you are emotional and seeking to escalate the conversation. Ask to postpone the conversation. You may find that you can't lower your defenses to give your partner a fair hearing. If you're in a heightened emotional state, you might ask to excuse yourself and resume later. After all, neither of you will get much out of a defensive back and forth. Say something politely to excuse yourself. For instance, "I'm really sorry Terry. I want to have this talk, but I'm not feeling up to it just now. Can we continue in an hour or two?" Be sure to affirm the importance of the conversation, too, e.g. "I know this is important and something you want to say. I just don't feel so calm right now. Let's try later, OK?" Take responsibility for initiating the conversation again once you've calmed down and approach your partner to let them know you are ready to continue. Postponing the conversation does not mean avoiding it. In fact, chronic postponing can be another defensive tactic: "You're going to bring that up again? I'm too busy to talk about it now."[5] EditListening to Criticism Try not to take it personally. Yes, this is a big challenge. How can you avoid taking your partner's criticisms personally, especially when they are about you and your behavior? Think of it this way: your partner isn't criticizing to attack or belittle you but speaking out with love to improve your relationship. Give your partner the benefit of the doubt.[6] Try to understand why you feel personally attacked. Is it because you feel like your partner is being unfair or exaggerating? Maybe you feel insecure or ashamed? Try also to think about why your partner would raise criticisms. Your romantic partner is probably not trying to make you look bad or shame you, but just wants to reach you. Honest criticism is about communication, love, and growth. Hold your tongue and listen. There are often times when you're being criticized that you want to snap back and explain yourself. Try to use self-control. Listen and, as said, don't interrupt. Interjecting every few seconds with "but…but" only makes it seem like you don't care and don't want to listen to what your partner says.[7] Hold your tongue if you have the urge to say "but…" If it helps, try actually biting down gently on your tongue or on your lower lip. If you must say something, ask questions that make you express what your partner is saying instead, like "I just want to get this straight, you don't think that I help around the house enough?" or "Do I understand correctly, you think we spend too much time with my parents?" Ask for concrete examples. You'll understand and digest your partner's honest criticism better if, instead of getting angry, you ask questions. Ask your partner to be more specific, to give examples, and to elaborate. This will help you to actually listen to the criticism and to show that you are interested in your partner's views.[8] Try saying something like, "What is it exactly that makes you feel I'm too emotionally distant, Alex?" or "Can you give me a specific example of when you felt I was being selfish?" Keep in mind that you're asking questions to understand your partner's criticism. You're not asking so that you can turn the criticism around or nitpick. This is another defensive tactic you should try to disarm. Resist the urge to counter-attack. Accepting your partner's criticism will take honesty and openness. You won't get anywhere if your immediate reaction is to lash out and criticize back. This, too, is a defensive tactic and will only escalate the situation, leading to frustration.[9] Try not to give in to the temptation to attack your partner, e.g. "You don't think I help out enough around here? I never see you cleaning the garage and yard!" or "You're being unreasonable. You do lots of things that annoy me, too!" Don't deflect the criticism by trying to justify or normalize it, either. For example: "I don't see what the problem is. My friend Jen goes out drinking every night of the week." EditEmpathizing with Your Partner Listen to your partner's words. Being able to accept honest criticism requires you to show empathy for your partner. This means to stand in your partner's shoes and try to see and feel things as he or she does, if only partially. To do this, try using the techniques of active listening as an aid.[10] First of all, focus your attention on what your partner is saying. You don't need to say or do anything. Just listen and allow your partner to talk. Listening means just that – no talking for a while. You shouldn't try to interject your opinion. Instead, send verbal and non-verbal signals that you're listening by things like nodding, acknowledging important points, or by saying things like, "Mmmhmmm," "Yes," and "I see." Suspend judgement. Empathy requires you to temporarily leave your own mental space and try to enter your partner's and suspend your own perspective and opinions. This is definitely a challenge. But, by doing it you'll be better able to focus on your partner's feelings and appreciate the honest criticism.[11] Suspending your judgement doesn't mean you have to accept your partner's viewpoint. You can decide to disagree. But, for now, it means you need to push away your own views, opinions, and reactions. Empathizing with your partner involves affirmation, too. Don't disregard your partner's criticisms by insisting that they are wrong or not important, like by saying "Well, that just isn't a big deal" or "Oh, get over it!" Keep in mind that the point of listening isn't always to offer a solution, either. Your partner has an honest criticism. There may be a solution in the future, but the key thing now is to hear your partner out. Repeat your partner's meaning. Repeating what your partner says to you, in your own words, will help you to engage with the criticism actively. Stay respectful, for one. Then, restate and reformulate your partner's points to be sure that you've understood correctly.[12] Say that your partner makes a point in your discussion. Try to repeat back what you've heard in different words. For instance, "OK, what I hear is that you think I'm being a bit selfish. Is that correct?" or "What I think you to mean is that you're frustrated by my emotional distance." You can add questions to expand the discussion, as well. For example, you might say, "What is it about my relationship with my mom that you find so frustrating?" This will help you both communicate further. Affirm your partner. Lastly, try to let your partner know that you have heard by signaling this verbally. Say that you've listened and understood the criticism and will give it serious thought. Even if you don't agree with what your partner has said, at least affirm that it is important. This way, you leave the door open for future discussions.[13] For example, you can say something along the lines of "I don't necessarily agree, Tasha, but I respect your point of view." Or, "Thank you for being honest with me, Juan. I'll consider what you've said." Try to summarize your partner's viewpoint, along with your own so that you are on the same page. For example, "I see, so when I don't change the toilet paper roll you feel like I'm being lazy and taking advantage, and to me it's just something I don't think about doing and don't see why it's important. Does that sum it up?" If you both agree on your positions, try to come up with a workable solution you can both stick to. For example, "Ok, I could see how my actions could come off as being lazy, how about I put up a reminder in the bathroom to put on a new roll when it gets low?" EditRelated wikiHows Deal With Criticism Accept Criticism Deal With Harsh Criticism Make Up with Your Partner After a Fight EditSources and Citations Cite error: <ref> tags exist, but no <references/> tag was found

How to Calculate Divergence and Curl Posted: 05 Sep 2016 01:00 AM PDT In vector calculus, divergence and curl are two important types of operators used on vector fields. Because vector fields are ubiquitous, these two operators are widely applicable to the physical sciences. EditSteps EditDivergence Understand what divergence is. Divergence is a measure of source or sink at a particular point. – In other words, how much is flowing into or out of a point. Hence, it is only defined for vector fields and outputs a scalar. Below is an example of a field with a positive divergence. The divergence is recognized by ${\displaystyle \operatorname {div} }$ or ${\displaystyle \nabla \cdot }$, where the dot signifies the similarity to taking a dot product. Take the dot product of the partial derivatives with the components of ${\displaystyle {\mathbf {F} }}$, then sum the results. In Cartesian coordinates, the divergence of a function ${\displaystyle {\mathbf {F} }=P{\mathbf {i} }+Q{\mathbf {j} }+R{\mathbf {k} }}$ is as follows. ${\displaystyle \nabla \cdot {\mathbf {F} }=\left({\frac {\partial }{\partial x}},{\frac {\partial }{\partial y}},{\frac {\partial }{\partial z}}\right)\cdot (P,Q,R)={\frac {\partial P}{\partial x}}+{\frac {\partial Q}{\partial y}}+{\frac {\partial R}{\partial z}}}$ EditExample Define the function: ${\displaystyle {\mathbf {F} }=(3x^{2}-5x^{2}y^{4}){\mathbf {i} }+(xy^{4}z^{2}-\sin(2x^{2}z^{3})){\mathbf {j} }+(5z^{2}+yz){\mathbf {k} }}$ Take the partials of ${\displaystyle \nabla }$ and multiply them with the respective components of ${\displaystyle {\mathbf {F} }}$ like you would a dot product. ${\displaystyle {\frac {\partial P}{\partial x}}=6x-10xy^{4}}$ ${\displaystyle {\frac {\partial Q}{\partial y}}=4xy^{3}z^{2}}$ ${\displaystyle {\frac {\partial R}{\partial z}}=10z+y}$ Sum the results. ${\displaystyle \nabla \cdot {\mathbf {F} }=6x-10xy^{4}+4xy^{3}z^{2}+y+10z}$ As you can see, we have mapped from a vector field to a scalar field. EditCurl Understand what curl is. The curl, defined for vector fields, is, intuitively, the amount of circulation at any point. The operator outputs another vector field. A whirlpool in real life consists of water acting like a vector field with a nonzero curl. Above is an example of a field with negative curl (because it's rotating clockwise). The curl is recognized by ${\displaystyle \operatorname {curl} }$ or ${\displaystyle \nabla \times }$, where the times symbol signifies the similarity of taking a cross product. Set up the determinant. ${\displaystyle \nabla \times {\mathbf {F} }={\begin{vmatrix}{\mathbf {i} }&{\mathbf {j} }&{\mathbf {k} }\\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\P&Q&R\end{vmatrix}}}$ Here, we see why the curl operator is denoted with a ${\displaystyle \nabla \times }$. – It is similar to how we would normally denote the cross product of two vectors. Find the determinant of the matrix. Below, we do it by cofactor expansion (expansion by minors). ${\displaystyle \nabla \times {\mathbf {F} }=\left({\frac {\partial R}{\partial y}}-{\frac {\partial Q}{\partial z}}\right){\mathbf {i} }-\left({\frac {\partial R}{\partial x}}-{\frac {\partial P}{\partial z}}\right){\mathbf {j} }+\left({\frac {\partial Q}{\partial x}}-{\frac {\partial P}{\partial y}}\right){\mathbf {k} }}$ EditExample Define the function: ${\displaystyle {\mathbf {F} }=(5x^{2}y^{2}-7xz^{3}){\mathbf {i} }+(4x-5xy-y^{4}){\mathbf {j} }+(xz+z^{2}){\mathbf {k} }}$ Set up the matrix. ${\displaystyle \nabla \times {\mathbf {F} }={\begin{vmatrix}{\mathbf {i} }&{\mathbf {j} }&{\mathbf {k} }\\{\frac {\partial }{\partial x}}&{\frac {\partial }{\partial y}}&{\frac {\partial }{\partial z}}\\P&Q&R\end{vmatrix}},}$ where ${\displaystyle P=5x^{2}y^{2}-7xz^{3},}$ ${\displaystyle Q=4x-5xy-y^{4},}$ and ${\displaystyle R=xz+z^{2}.}$ Evaluate the determinant. ${\displaystyle \left({\frac {\partial R}{\partial y}}-{\frac {\partial Q}{\partial z}}\right){\mathbf {i} }=0-0}$ ${\displaystyle \left({\frac {\partial R}{\partial x}}-{\frac {\partial P}{\partial z}}\right){\mathbf {j} }=z-(-21xz^{2})}$ ${\displaystyle \left({\frac {\partial Q}{\partial x}}-{\frac {\partial P}{\partial y}}\right){\mathbf {k} }=(4-5y)-10x^{2}{y}}$ Arrive at the solution. ${\displaystyle \nabla \times {\mathbf {F} }=-(z+21xz^{2}){\mathbf {j} }+(4-5y-10x^{2}y){\mathbf {k} }}$ Note that we have mapped to another vector field. EditTips There are several identities that are useful and worth memorizing. Here is a partial list of the more important ones, the second derivatives. You can prove them by direct computation. ${\displaystyle \nabla \cdot (\nabla \times {\mathbf {F} })=0}$ ${\displaystyle \nabla \times (\nabla f)=0}$ The Laplacian is a compound operator and is defined as the divergence of the gradient ${\displaystyle \nabla \cdot \nabla =\nabla ^{2}.}$ We write out the Laplacian in Cartesian coordinates below. The operator is especially useful in physics and engineering. For example, the potential of an electrostatic field can be described using Poisson's equation, which involves the Laplacian. ${\displaystyle \nabla ^{2}\phi ={\frac {\partial ^{2}\phi }{\partial x^{2}}}+{\frac {\partial ^{2}\phi }{\partial y^{2}}}+{\frac {\partial ^{2}\phi }{\partial z^{2}}}}$

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